magma: G := SmallGroup(256, 55636);
gap: G := SmallGroup(256, 55636);
sage: G = libgap.SmallGroup(256, 55636)
sage: # This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
G = gap.new('PcGroupCode(578721485682639056,256)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.7;
Group information
Description: $C_4^2.C_2^4$ Order: \(256\)\(\medspace = 2^{8} \)
magma: Order(G);
gap: Order(G);
sage: G.order()
sage: G.Order()
Exponent: \(4\)\(\medspace = 2^{2} \)
magma: Exponent(G);
gap: Exponent(G);
sage: G.exponent()
sage: G.Exponent()
Automorphism group :$C_2^9.A_4.C_2^6.C_2^5$ , of order \(12582912\)\(\medspace = 2^{22} \cdot 3 \)
gap: AutomorphismGroup(G);
magma: AutomorphismGroup(G);
sage: G.AutomorphismGroup()
Composition factors :$C_2$ x 8
magma: CompositionFactors(G);
gap: CompositionSeries(G);
sage: G.composition_series()
sage: G.CompositionSeries()
Nilpotency class: $2$
magma: NilpotencyClass(G);
gap: NilpotencyClassOfGroup(G);
sage: G.NilpotencyClassOfGroup()
Derived length: $2$
magma: DerivedLength(G);
gap: DerivedLength(G);
sage: G.DerivedLength()
This group is nonabelian , a $p$-group (hence nilpotent , solvable , supersolvable , monomial , elementary , and hyperelementary ), and metabelian .
magma: IsAbelian(G);
gap: IsAbelian(G);
sage: G.is_abelian()
sage: G.IsAbelian()
magma: IsCyclic(G);
gap: IsCyclic(G);
sage: G.is_cyclic()
sage: G.IsCyclic()
magma: IsNilpotent(G);
gap: IsNilpotentGroup(G);
sage: G.is_nilpotent()
sage: G.IsNilpotentGroup()
magma: IsSolvable(G);
gap: IsSolvableGroup(G);
sage: G.is_solvable()
sage: G.IsSolvableGroup()
gap: IsSupersolvableGroup(G);
sage: G.is_supersolvable()
sage: G.IsSupersolvableGroup()
magma: IsSimple(G);
gap: IsSimpleGroup(G);
sage: G.IsSimpleGroup()
Group statistics
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))
magma: // Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i
CharacterDegrees(G);
gap: # Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
CharacterDegrees(G);
sage: # Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
character_degrees = [c[0] for c in G.character_table()]
[[n, character_degrees.count(n)] for n in set(character_degrees)]
sage: G.CharacterDegrees()
Minimal presentations
Over $\mathbb{C}$
Over $\mathbb{R}$
Over $\mathbb{Q}$
Irreducible none
none
none
Arbitrary not computed
not computed
not computed
Constructions
Presentation :
${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{4}=f^{4}=[a,c]= \!\cdots\! \rangle}$
magma: G := PCGroup([8, 2, 2, 2, 2, 2, 2, 2, 2, 2081, 748, 116, 166]); a,b,c,d,e,f := Explode([G.1, G.2, G.3, G.4, G.5, G.7]); AssignNames(~G, ["a", "b", "c", "d", "e", "e2", "f", "f2"]);
gap: G := PcGroupCode(578721485682639056,256); a := G.1; b := G.2; c := G.3; d := G.4; e := G.5; f := G.7;
sage: # This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
G = gap.new('PcGroupCode(578721485682639056,256)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.7;
sage: # This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
G = gap.new('PcGroupCode(578721485682639056,256)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.7;
Direct product :$C_2$ $\, \times\, $ $(C_4^2.C_2^3)$
Semidirect product :$(C_4\times Q_8)$ $\,\rtimes\,$ $C_2^3$ $(D_4:C_2^3)$ $\,\rtimes\,$ $C_4$ $(C_4\times D_4)$ $\,\rtimes\,$ $C_2^3$ $(C_2^3.D_4)$ $\,\rtimes\,$ $C_2^2$ all 28
Trans. wreath product :not isomorphic to a non-trivial transitive wreath product
Non-split product :$C_2^5$ . $C_2^3$ $C_2^4$ . $C_2^4$ (3)$C_4^2$ . $C_2^4$ (2)$C_2^3$ . $C_2^5$ (3)all 39
Elements of the group are displayed as words in the presentation generators from the presentation above.
Homology
Subgroups
magma: Subgroups(G);
gap: AllSubgroups(G);
sage: G.subgroups()
sage: G.AllSubgroups()
There are 6799 subgroups in 5767 conjugacy classes , 5287 normal (11 characteristic ).
Characteristic subgroups are shown in this color . Normal (but not characteristic) subgroups are shown in this color .
Special subgroups
Center :$Z \simeq$ $C_2^2\times C_4$ $G/Z \simeq$ $C_2^4$
magma: Center(G);
gap: Center(G);
sage: G.center()
sage: G.Center()
Commutator :$G' \simeq$ $C_2$ $G/G' \simeq$ $C_2^5\times C_4$
magma: CommutatorSubgroup(G);
gap: DerivedSubgroup(G);
sage: G.commutator()
sage: G.DerivedSubgroup()
Frattini :$\Phi \simeq$ $C_2^2$ $G/\Phi \simeq$ $C_2^6$
magma: FrattiniSubgroup(G);
gap: FrattiniSubgroup(G);
sage: G.frattini_subgroup()
sage: G.FrattiniSubgroup()
Fitting :$\operatorname{Fit} \simeq$ $C_4^2.C_2^4$ $G/\operatorname{Fit} \simeq$ $C_1$
magma: FittingSubgroup(G);
gap: FittingSubgroup(G);
sage: G.fitting_subgroup()
sage: G.FittingSubgroup()
Radical :$R \simeq$ $C_4^2.C_2^4$ $G/R \simeq$ $C_1$
magma: Radical(G);
gap: SolvableRadical(G);
sage: G.SolvableRadical()
Socle :$\operatorname{soc} \simeq$ $C_2^3$ $G/\operatorname{soc} \simeq$ $C_2^5$
magma: Socle(G);
gap: Socle(G);
sage: G.socle()
sage: G.Socle()
2-Sylow subgroup :
$P_{ 2 } \simeq$ $C_4^2.C_2^4$
Hi
diagram
profile
all subgroups
normal subgroups
up to conjugacy
up to automorphism
Classes of subgroups up to conjugation
No diagram available: subgroups only stored up to automorphism
Classes of subgroups up to automorphism
No diagram available
Normal subgroups
No diagram available: subgroups only stored up to automorphism
Normal subgroups up to automorphism
No diagram available
Classes of subgroups up to conjugation
Order 256: $C_4^2.C_2^4$ Order 128: $C_4^2.C_2^3$ x 32, $C_4^2.C_2^3$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2:C_2^3$ x 6, $C_4^2:C_2^3$ x 6, $C_4^2.C_2^3$ x 2, $D_4:C_2^4$ Order 64: $C_4^2:C_2^2$ x 136, $C_4^2.C_2^2$ x 128, $C_4^2.C_2^2$ x 128, $C_4^2:C_2^2$ x 96, $D_4:C_2^3$ x 56, $C_4^2:C_2^2$ x 48, $C_4^2.C_2^2$ x 32, $C_4^2.C_2^2$ x 16, $C_2^3.D_4$ x 12, $C_2^4:C_4$ x 12, $C_2^2\times C_4^2$ x 12, $D_4\times C_2^3$ x 3, $C_2^4\times C_4$ x 3, $Q_8\times C_2^3$ Order 32: $D_4:C_2^2$ x 448, $C_4\times D_4$ x 384, $C_4^2:C_2$ x 320, $C_4\times Q_8$ x 128, $C_2^2.D_4$ x 128, $C_2^3:C_4$ x 128, $C_2\times C_4^2$ x 128, $C_2^3\times C_4$ x 112, $C_2^2\times D_4$ x 84, $C_2^2\times Q_8$ x 28, $C_2^5$ x 3 Order 16: $C_2^2\times C_4$ x 552, $D_4:C_2$ x 512, $C_2\times D_4$ x 336, $C_4:C_4$ x 128, $C_2^2:C_4$ x 128, $C_4^2$ x 128, $C_2\times Q_8$ x 112, $C_2^4$ x 55 Order 8: $C_2\times C_4$ x 464, $D_4$ x 192, $C_2^3$ x 191, $Q_8$ x 64 Order 4: $C_2^2$ x 167, $C_4$ x 64 Order 2: $C_2$ x 35 Order 1: $C_1$
Classes of subgroups up to automorphism
Order 256: $C_4^2.C_2^4$ Order 128: $D_4:C_2^4$ , $C_4^2.C_2^3$ , $C_4^2.C_2^3$ , $C_4^2.C_2^3$ , $C_4^2:C_2^3$ , $C_4^2.C_2^3$ , $C_4^2:C_2^3$ Order 64: $C_4^2:C_2^2$ x 5, $D_4:C_2^3$ x 4, $Q_8\times C_2^3$ , $D_4\times C_2^3$ , $C_2^4\times C_4$ , $C_4^2.C_2^2$ , $C_4^2.C_2^2$ , $C_4^2:C_2^2$ , $C_4^2.C_2^2$ , $C_4^2.C_2^2$ , $C_4^2:C_2^2$ , $C_2^3.D_4$ , $C_2^4:C_4$ , $C_2^2\times C_4^2$ Order 32: $C_2^3\times C_4$ x 7, $D_4:C_2^2$ x 5, $C_4^2:C_2$ x 4, $C_2^2\times Q_8$ x 3, $C_2^2\times D_4$ x 3, $C_2^2.D_4$ x 3, $C_2^3:C_4$ x 3, $C_2\times C_4^2$ x 3, $C_2^5$ , $C_4\times Q_8$ , $C_4\times D_4$ Order 16: $C_2^2\times C_4$ x 15, $C_2^4$ x 5, $C_2\times Q_8$ x 3, $C_2\times D_4$ x 3, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $D_4:C_2$ x 2 Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 10, $Q_8$ , $D_4$ Order 4: $C_2^2$ x 10, $C_4$ x 4 Order 2: $C_2$ x 5 Order 1: $C_1$
Normal subgroups (quotient in parentheses)
Order 256: $C_4^2.C_2^4$ ($C_1$) Order 128: $C_4^2.C_2^3$ ($C_2$) x 32, $C_4^2.C_2^3$ ($C_2$) x 8, $C_4^2.C_2^3$ ($C_2$) x 8, $C_4^2:C_2^3$ ($C_2$) x 6, $C_4^2:C_2^3$ ($C_2$) x 6, $C_4^2.C_2^3$ ($C_2$) x 2, $D_4:C_2^4$ ($C_2$) Order 64: $C_4^2:C_2^2$ ($C_2^2$) x 136, $C_4^2.C_2^2$ ($C_2^2$) x 128, $C_4^2.C_2^2$ ($C_2^2$) x 128, $C_4^2:C_2^2$ ($C_2^2$) x 96, $C_4^2:C_2^2$ ($C_2^2$) x 48, $D_4:C_2^3$ ($C_4$) x 32, $C_4^2.C_2^2$ ($C_2^2$) x 32, $D_4:C_2^3$ ($C_2^2$) x 24, $C_4^2.C_2^2$ ($C_2^2$) x 16, $C_2^3.D_4$ ($C_2^2$) x 12, $C_2^4:C_4$ ($C_2^2$) x 12, $C_2^2\times C_4^2$ ($C_2^2$) x 12, $D_4\times C_2^3$ ($C_2^2$) x 3, $C_2^4\times C_4$ ($C_2^2$) x 3, $Q_8\times C_2^3$ ($C_2^2$) Order 32: $D_4:C_2^2$ ($C_2\times C_4$) x 384, $C_4\times D_4$ ($C_2^3$) x 384, $C_4^2:C_2$ ($C_2^3$) x 320, $C_4\times Q_8$ ($C_2^3$) x 128, $C_2^2.D_4$ ($C_2^3$) x 128, $C_2^3:C_4$ ($C_2^3$) x 128, $C_2\times C_4^2$ ($C_2^3$) x 128, $D_4:C_2^2$ ($C_2^3$) x 64, $C_2^3\times C_4$ ($C_2^3$) x 64, $C_2^2\times D_4$ ($C_2\times C_4$) x 48, $C_2^3\times C_4$ ($C_2\times C_4$) x 48, $C_2^2\times D_4$ ($C_2^3$) x 36, $C_2^2\times Q_8$ ($C_2\times C_4$) x 16, $C_2^2\times Q_8$ ($C_2^3$) x 12, $C_2^5$ ($C_2^3$) x 3 Order 16: $D_4:C_2$ ($C_2^2\times C_4$) x 512, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 320, $C_2\times D_4$ ($C_2^2\times C_4$) x 288, $C_2^2\times C_4$ ($C_2^4$) x 184, $C_4:C_4$ ($C_2^4$) x 128, $C_2^2:C_4$ ($C_2^4$) x 128, $C_4^2$ ($C_2^4$) x 128, $C_2\times Q_8$ ($C_2^2\times C_4$) x 96, $C_2\times D_4$ ($C_2^4$) x 48, $C_2^4$ ($C_2^2\times C_4$) x 24, $C_2^4$ ($C_2^4$) x 19, $C_2\times Q_8$ ($C_2^4$) x 16 Order 8: $C_2\times C_4$ ($C_2^3\times C_4$) x 288, $D_4$ ($C_2^3\times C_4$) x 192, $C_2^3$ ($C_2^3\times C_4$) x 76, $Q_8$ ($C_2^3\times C_4$) x 64, $C_2\times C_4$ ($C_2^5$) x 48, $C_2^3$ ($C_2^5$) x 15 Order 4: $C_4$ ($C_2^4\times C_4$) x 32, $C_2^2$ ($C_2^4\times C_4$) x 30, $C_2^2$ ($C_4.C_2^4$) x 4, $C_2^2$ ($C_2^6$) Order 2: $C_2$ ($C_4^2.C_2^3$) x 4, $C_2$ ($C_4.C_2^5$) x 2, $C_2$ ($C_2^5\times C_4$) Order 1: $C_1$ ($C_4^2.C_2^4$)
Normal subgroups up to automorphism (quotient in parentheses)
Order 256: $C_4^2.C_2^4$ ($C_1$) Order 128: $D_4:C_2^4$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2:C_2^3$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_4^2:C_2^3$ ($C_2$) Order 64: $C_4^2:C_2^2$ ($C_2^2$) x 5, $D_4:C_2^3$ ($C_2^2$) x 3, $D_4:C_2^3$ ($C_4$), $Q_8\times C_2^3$ ($C_2^2$), $D_4\times C_2^3$ ($C_2^2$), $C_2^4\times C_4$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^4:C_4$ ($C_2^2$), $C_2^2\times C_4^2$ ($C_2^2$) Order 32: $C_2^3\times C_4$ ($C_2^3$) x 6, $C_4^2:C_2$ ($C_2^3$) x 4, $D_4:C_2^2$ ($C_2\times C_4$) x 3, $C_2^2.D_4$ ($C_2^3$) x 3, $C_2^3:C_4$ ($C_2^3$) x 3, $C_2\times C_4^2$ ($C_2^3$) x 3, $D_4:C_2^2$ ($C_2^3$) x 2, $C_2^2\times Q_8$ ($C_2^3$) x 2, $C_2^2\times D_4$ ($C_2^3$) x 2, $C_2^5$ ($C_2^3$), $C_2^2\times Q_8$ ($C_2\times C_4$), $C_2^2\times D_4$ ($C_2\times C_4$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_4\times Q_8$ ($C_2^3$), $C_4\times D_4$ ($C_2^3$) Order 16: $C_2^2\times C_4$ ($C_2^4$) x 9, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 5, $C_2^4$ ($C_2^4$) x 3, $C_4:C_4$ ($C_2^4$) x 2, $C_2^2:C_4$ ($C_2^4$) x 2, $C_4^2$ ($C_2^4$) x 2, $D_4:C_2$ ($C_2^2\times C_4$) x 2, $C_2\times Q_8$ ($C_2^2\times C_4$) x 2, $C_2\times D_4$ ($C_2^2\times C_4$) x 2, $C_2^4$ ($C_2^2\times C_4$), $C_2\times Q_8$ ($C_2^4$), $C_2\times D_4$ ($C_2^4$) Order 8: $C_2\times C_4$ ($C_2^3\times C_4$) x 7, $C_2\times C_4$ ($C_2^5$) x 4, $C_2^3$ ($C_2^5$) x 3, $C_2^3$ ($C_2^3\times C_4$) x 3, $Q_8$ ($C_2^3\times C_4$), $D_4$ ($C_2^3\times C_4$) Order 4: $C_2^2$ ($C_2^4\times C_4$) x 3, $C_4$ ($C_2^4\times C_4$) x 3, $C_2^2$ ($C_2^6$), $C_2^2$ ($C_4.C_2^4$) Order 2: $C_2$ ($C_4.C_2^5$), $C_2$ ($C_2^5\times C_4$), $C_2$ ($C_4^2.C_2^3$) Order 1: $C_1$ ($C_4^2.C_2^4$)
Series
Derived series
$C_4^2.C_2^4$
$\rhd$
$C_2$
$\rhd$
$C_1$
magma: DerivedSeries(G);
gap: DerivedSeriesOfGroup(G);
sage: G.derived_series()
sage: G.DerivedSeriesOfGroup()
Chief series
$C_4^2.C_2^4$
$\rhd$
$C_4^2:C_2^3$
$\rhd$
$C_4^2:C_2^2$
$\rhd$
$C_4^2:C_2$
$\rhd$
$C_4^2$
$\rhd$
$C_2\times C_4$
$\rhd$
$C_2^2$
$\rhd$
$C_2$
$\rhd$
$C_1$
magma: ChiefSeries(G);
gap: ChiefSeries(G);
sage: G.ChiefSeries()
Lower central series
$C_4^2.C_2^4$
$\rhd$
$C_2$
$\rhd$
$C_1$
magma: LowerCentralSeries(G);
gap: LowerCentralSeriesOfGroup(G);
sage: G.lower_central_series()
sage: G.LowerCentralSeriesOfGroup()
Upper central series
$C_1$
$\lhd$
$C_2^2\times C_4$
$\lhd$
$C_4^2.C_2^4$
magma: UpperCentralSeries(G);
gap: UpperCentralSeriesOfGroup(G);
sage: G.upper_central_series()
sage: G.UpperCentralSeriesOfGroup()
Character theory
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: G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
See the $136 \times 136$ character table (warning: may be slow to load).
Alternatively, you may search for characters of this group with desired properties.
See the $100 \times 100$ rational character table .
