LMFDB - Abstract group 94248.a: $C_{11781}:Q

comment:Define the group as a permutation group

magma:G := PermutationGroup< 52 | (2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51) >;

gap:G := Group( (2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51) );

sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)', '(1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(11816649444987855719426444033694037767236423670581839581524961504606992106753430943,94248)'); a = G.1; b = G.2;

Group information

Description:	$C_{11781}:Q_8$
Order:	$94248$$\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$47124$$\medspace = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	Group of order $3548160$$\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$, $C_{11}$, $C_{17}$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$2$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

Order	1	2	3	4	6	7	9	11	12	14	17	18	21	22	28	33	34	36	42	44	51	63	66	68	77	84	99	102	119	126	132	153	154	187	198	204	231	238	252	306	308	357	374	396	462	476	561	612	693	714	748	924	1071	1122	1309	1386	1428	1683	2142	2244	2618	2772	3366	3927	4284	5236	6732	7854	11781	15708	23562	47124
Elements	1	1	2	310	2	6	6	10	620	6	16	6	12	10	12	20	16	1860	12	20	32	36	20	4960	60	24	60	32	96	36	40	96	60	160	60	9920	120	96	72	96	120	192	160	120	120	192	320	29760	360	192	320	240	576	320	960	360	384	960	576	640	960	720	960	1920	1152	1920	1920	1920	5760	3840	5760	11520	94248
Conjugacy classes	1	1	2	3	2	3	6	5	6	3	16	6	6	5	6	10	16	18	6	10	32	18	10	48	30	12	30	32	48	18	20	96	30	80	30	96	60	48	36	96	60	96	80	60	60	96	160	288	180	96	160	120	288	160	480	180	192	480	288	320	480	360	480	960	576	960	960	960	2880	1920	2880	5760	24021
Divisions	1	1	1	3	1	1	1	1	3	1	1	1	1	1	1	1	1	3	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	84
Autjugacy classes	1	1	1	2	1	1	1	1	2	1	1	1	1	1	1	1	1	2	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	78

Minimal presentations

Permutation degree:	$52$
Transitive degree:	$94248$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	$\langle a, b \mid b^{47124}=1, a^{2}=b^{2002}, b^{a}=b^{307} \rangle$ < a, b \| b^47124, b^2002a^-2, b^307a^-1b^-1a >
comment:Define the group with the given generators and relations magma:G := PCGroup([8, -2, -2, -2, -3, -3, -7, -11, -17, 32032, 9825, 41, 29474, 66, 78595, 123, 294724, 156, 1060997, 381, 3386886, 670]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b4", "b12", "b36", "b252", "b2772"]); gap:G := PcGroupCode(11816649444987855719426444033694037767236423670581839581524961504606992106753430943,94248); a := G.1; b := G.2; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(11816649444987855719426444033694037767236423670581839581524961504606992106753430943,94248)'); a = G.1; b = G.2; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(11816649444987855719426444033694037767236423670581839581524961504606992106753430943,94248)'); a = G.1; b = G.2;
Permutation group:	Degree $52$ $\langle(2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27) \!\cdots\! \rangle$ (2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51)
comment:Define the group as a permutation group magma:G := PermutationGroup< 52 \| (2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51) >; gap:G := Group( (2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51) ); sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(8,9,11,14)(10,13,15,12)(16,17,18,19,20,21,22,23,24)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)', '(1,2,4,6,7,5,3)(8,10,11,15)(9,12,14,13)(16,17,18,19,20,21,22,23,24)(25,26,28,30,32,34,35,33,31,29,27)(36,38,40,42,44,46,48,50,52,37,39,41,43,45,47,49,51)'])
Matrix group:	$\left\langle \left(\begin{array}{rr} 65 & 216 \\ 141 & 242 \end{array}\right), \left(\begin{array}{rr} 155 & 258 \\ 113 & 155 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{307})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 2, GF(307) \| [[65, 216, 141, 242], [155, 258, 113, 155]] >; gap:G := Group([[[ Z(307)^254, Z(307)^216 ], [ Z(307)^62, Z(307)^101 ]], [[ Z(307)^294, Z(307)^79 ], [ Z(307)^78, Z(307)^294 ]]]); sage:MS = MatrixSpace(GF(307), 2, 2) G = MatrixGroup([MS([[65, 216], [141, 242]]), MS([[155, 258], [113, 155]])])
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{6732}$ . $D_7$	$C_{612}$ . $D_{77}$	$C_{4284}$ . $D_{11}$	$C_{3366}$ . $D_{14}$	all 64

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{306} \simeq C_{2}^{2} \times C_{9} \times C_{17}$	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_1$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	not computed	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

There are 1800 subgroups in 144 conjugacy classes, 90 normal (78 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{306}$	$G/Z \simeq$ $D_{154}$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_{154}$	$G/G' \simeq$ $C_2\times C_{306}$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_{102}\times D_{77}$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_{47124}$	$G/\operatorname{Fit} \simeq$ $C_2$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_{11781}:Q_8$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_{7854}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Subgroup diagram and profile

Normal subgroups (quotient in parentheses)

Order 94248: $C_{11781}:Q_8$ ($C_1$)

Order 47124: $C_{77}:C_{612}$ ($C_2$) x 2, $C_{47124}$ ($C_2$)

Order 31416: $C_{3927}:Q_8$ ($C_3$)

Order 23562: $C_{23562}$ ($C_2^2$)

Order 15708: $C_{77}:C_{204}$ ($C_6$) x 2, $C_{15708}$ ($C_6$)

Order 11781: $C_{11781}$ ($Q_8$)

Order 10472: $C_{1309}:Q_8$ ($C_9$)

Order 7854: $C_{7854}$ ($C_2\times C_6$)

Order 6732: $C_{6732}$ ($D_7$)

Order 5544: $C_{693}:Q_8$ ($C_{17}$)

Order 5236: $C_{17}\times C_{77}:C_4$ ($C_{18}$) x 2, $C_{5236}$ ($C_{18}$)

Order 4284: $C_{4284}$ ($D_{11}$)

Order 3927: $C_{3927}$ ($C_3\times Q_8$)

Order 3366: $C_{3366}$ ($D_{14}$)

Order 2772: $C_{693}:C_4$ ($C_{34}$) x 2, $C_{2772}$ ($C_{34}$)

Order 2618: $C_{2618}$ ($C_2\times C_{18}$)

Order 2244: $C_{2244}$ ($C_3\times D_7$)

Order 2142: $C_{2142}$ ($D_{22}$)

Order 1848: $C_{231}:Q_8$ ($C_{51}$)

Order 1683: $C_{1683}$ ($C_7:Q_8$)

Order 1428: $C_{1428}$ ($C_3\times D_{11}$)

Order 1386: $C_{1386}$ ($C_2\times C_{34}$)

Order 1309: $C_{1309}$ ($Q_8\times C_9$)

Order 1122: $C_{1122}$ ($C_3\times D_{14}$)

Order 1071: $C_{1071}$ ($C_{11}:Q_8$)

Order 924: $C_{231}:C_4$ ($C_{102}$) x 2, $C_{924}$ ($C_{102}$)

Order 748: $C_{748}$ ($C_9\times D_7$)

Order 714: $C_{714}$ ($C_3\times D_{22}$)

Order 693: $C_{693}$ ($Q_8\times C_{17}$)

Order 616: $C_{77}:Q_8$ ($C_{153}$)

Order 612: $C_{612}$ ($D_{77}$)

Order 561: $C_{561}$ ($C_{21}:Q_8$)

Order 476: $C_{476}$ ($C_9\times D_{11}$)

Order 462: $C_{462}$ ($C_2\times C_{102}$)

Order 396: $C_{396}$ ($D_7\times C_{17}$)

Order 374: $C_{374}$ ($C_9\times D_{14}$)

Order 357: $C_{357}$ ($C_{33}:Q_8$)

Order 308: $C_{77}:C_4$ ($C_{306}$) x 2, $C_{308}$ ($C_{306}$)

Order 306: $C_{306}$ ($D_{154}$)

Order 252: $C_{252}$ ($C_{17}\times D_{11}$)

Order 238: $C_{238}$ ($C_9\times D_{22}$)

Order 231: $C_{231}$ ($Q_8\times C_{51}$)

Order 204: $C_{204}$ ($C_3\times D_{77}$)

Order 198: $C_{198}$ ($C_{17}\times D_{14}$)

Order 187: $C_{187}$ ($C_{63}:Q_8$)

Order 154: $C_{154}$ ($C_2\times C_{306}$)

Order 153: $C_{153}$ ($C_{77}:Q_8$)

Order 132: $C_{132}$ ($D_7\times C_{51}$)

Order 126: $C_{126}$ ($C_{17}\times D_{22}$)

Order 119: $C_{119}$ ($C_{99}:Q_8$)

Order 102: $C_{102}$ ($C_3\times D_{154}$)

Order 99: $C_{99}$ ($C_{119}:Q_8$)

Order 84: $C_{84}$ ($D_{11}\times C_{51}$)

Order 77: $C_{77}$ ($Q_8\times C_{153}$)

Order 68: $C_{68}$ ($C_9\times D_{77}$)

Order 66: $C_{66}$ ($D_7\times C_{102}$)

Order 63: $C_{63}$ ($C_{187}:Q_8$)

Order 51: $C_{51}$ ($C_{231}:Q_8$)

Order 44: $C_{44}$ ($D_7\times C_{153}$)

Order 42: $C_{42}$ ($D_{11}\times C_{102}$)

Order 36: $C_{36}$ ($C_{17}\times D_{77}$)

Order 34: $C_{34}$ ($C_9\times D_{154}$)

Order 33: $C_{33}$ ($C_{357}:Q_8$)

Order 28: $C_{28}$ ($D_{11}\times C_{153}$)

Order 22: $C_{22}$ ($D_7\times C_{306}$)

Order 21: $C_{21}$ ($C_{561}:Q_8$)

Order 18: $C_{18}$ ($C_{17}\times D_{154}$)

Order 17: $C_{17}$ ($C_{693}:Q_8$)

Order 14: $C_{14}$ ($D_{11}\times C_{306}$)

Order 12: $C_{12}$ ($C_{51}\times D_{77}$)

Order 11: $C_{11}$ ($C_{1071}:Q_8$)

Order 9: $C_9$ ($C_{1309}:Q_8$)

Order 7: $C_7$ ($C_{1683}:Q_8$)

Order 6: $C_6$ ($C_{102}\times D_{77}$)

Order 4: $C_4$ ($C_{153}\times D_{77}$)

Order 3: $C_3$ ($C_{3927}:Q_8$)

Order 2: $C_2$ ($C_{153}\times D_{154}$)

Order 1: $C_1$ ($C_{11781}:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 94248: $C_{11781}:Q_8$ ($C_1$)

Order 47124: $C_{47124}$ ($C_2$)

Order 31416: $C_{3927}:Q_8$ ($C_3$)

Order 23562: $C_{23562}$ ($C_2^2$)

Order 11781: $C_{11781}$ ($Q_8$)

Order 10472: $C_{1309}:Q_8$ ($C_9$)

Order 7854: $C_{7854}$ ($C_2\times C_6$)

Order 5544: $C_{693}:Q_8$ ($C_{17}$)

Order 3927: $C_{3927}$ ($C_3\times Q_8$)

Order 3366: $C_{3366}$ ($D_{14}$)

Order 2618: $C_{2618}$ ($C_2\times C_{18}$)

Order 2142: $C_{2142}$ ($D_{22}$)

Order 1848: $C_{231}:Q_8$ ($C_{51}$)

Order 1683: $C_{1683}$ ($C_7:Q_8$)

Order 1386: $C_{1386}$ ($C_2\times C_{34}$)

Order 1309: $C_{1309}$ ($Q_8\times C_9$)

Order 1122: $C_{1122}$ ($C_3\times D_{14}$)

Order 1071: $C_{1071}$ ($C_{11}:Q_8$)

Order 714: $C_{714}$ ($C_3\times D_{22}$)

Order 693: $C_{693}$ ($Q_8\times C_{17}$)

Order 616: $C_{77}:Q_8$ ($C_{153}$)

Order 561: $C_{561}$ ($C_{21}:Q_8$)

Order 462: $C_{462}$ ($C_2\times C_{102}$)

Order 374: $C_{374}$ ($C_9\times D_{14}$)

Order 357: $C_{357}$ ($C_{33}:Q_8$)

Order 306: $C_{306}$ ($D_{154}$)

Order 238: $C_{238}$ ($C_9\times D_{22}$)

Order 231: $C_{231}$ ($Q_8\times C_{51}$)

Order 198: $C_{198}$ ($C_{17}\times D_{14}$)

Order 187: $C_{187}$ ($C_{63}:Q_8$)

Order 154: $C_{154}$ ($C_2\times C_{306}$)

Order 153: $C_{153}$ ($C_{77}:Q_8$)

Order 126: $C_{126}$ ($C_{17}\times D_{22}$)

Order 119: $C_{119}$ ($C_{99}:Q_8$)

Order 102: $C_{102}$ ($C_3\times D_{154}$)

Order 99: $C_{99}$ ($C_{119}:Q_8$)

Order 77: $C_{77}$ ($Q_8\times C_{153}$)

Order 66: $C_{66}$ ($D_7\times C_{102}$)

Order 63: $C_{63}$ ($C_{187}:Q_8$)

Order 51: $C_{51}$ ($C_{231}:Q_8$)

Order 42: $C_{42}$ ($D_{11}\times C_{102}$)

Order 34: $C_{34}$ ($C_9\times D_{154}$)

Order 33: $C_{33}$ ($C_{357}:Q_8$)

Order 22: $C_{22}$ ($D_7\times C_{306}$)

Order 21: $C_{21}$ ($C_{561}:Q_8$)

Order 18: $C_{18}$ ($C_{17}\times D_{154}$)

Order 17: $C_{17}$ ($C_{693}:Q_8$)

Order 14: $C_{14}$ ($D_{11}\times C_{306}$)

Order 11: $C_{11}$ ($C_{1071}:Q_8$)

Order 9: $C_9$ ($C_{1309}:Q_8$)

Order 7: $C_7$ ($C_{1683}:Q_8$)

Order 6: $C_6$ ($C_{102}\times D_{77}$)

Order 3: $C_3$ ($C_{3927}:Q_8$)

Order 2: $C_2$ ($C_{153}\times D_{154}$)

Order 1: $C_1$ ($C_{11781}:Q_8$)

Series

Derived series

$C_{11781}:Q_8$

$\rhd$

$C_{154}$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_{11781}:Q_8$

$\rhd$

$C_{77}:C_{612}$

$\rhd$

$C_{23562}$

$\rhd$

$C_{11781}$

$\rhd$

$C_{3927}$

$\rhd$

$C_{1309}$

$\rhd$

$C_{187}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_{11781}:Q_8$

$\rhd$

$C_{154}$

$\rhd$

$C_{77}$

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

Upper central series

$C_1$

$\lhd$

$C_{306}$

$\lhd$

$C_{612}$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

The $24021 \times 24021$ character table is not available for this group.

Rational character table

The $84 \times 84$ rational character table is not available for this group.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	94248.a
Order	\( 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
Exponent	\( 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{2} \cdot 3^{2} \cdot 17 \)
$\card{Z(G)}$	\( 2 \cdot 3^{2} \cdot 17 \)
$\card{\Aut(G)}$	\( 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
$\card{\mathrm{Out}(G)}$	\( 2^{8} \cdot 3^{2} \cdot 5 \)
Perm deg.	$52$
Trans deg.	$94248$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Group information

Group statistics

Minimal presentations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.