magma: G := SymmetricGroup(8);
gap: G := SymmetricGroup(8);
sage: G = SymmetricGroup(8)
Group information
Description: $S_8$ Order: \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
magma: Order(G);
gap: Order(G);
sage: G.order()
sage_gap: G.Order()
Exponent: \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
magma: Exponent(G);
gap: Exponent(G);
sage: G.exponent()
sage_gap: G.Exponent()
Automorphism group :$S_8$ , of order \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
gap: AutomorphismGroup(G);
magma: AutomorphismGroup(G);
sage_gap: G.AutomorphismGroup()
Composition factors :$C_2$ , $A_8$
magma: CompositionFactors(G);
gap: CompositionSeries(G);
sage: G.composition_series()
sage_gap: G.CompositionSeries()
Derived length: $1$
magma: DerivedLength(G);
gap: DerivedLength(G);
sage_gap: G.DerivedLength()
This group is nonabelian , almost simple , nonsolvable , and rational .
magma: IsAbelian(G);
gap: IsAbelian(G);
sage: G.is_abelian()
sage_gap: G.IsAbelian()
magma: IsCyclic(G);
gap: IsCyclic(G);
sage: G.is_cyclic()
sage_gap: G.IsCyclic()
magma: IsNilpotent(G);
gap: IsNilpotentGroup(G);
sage: G.is_nilpotent()
sage_gap: G.IsNilpotentGroup()
magma: IsSolvable(G);
gap: IsSolvableGroup(G);
sage: G.is_solvable()
sage_gap: G.IsSolvableGroup()
gap: IsSupersolvableGroup(G);
sage: G.is_supersolvable()
sage_gap: G.IsSupersolvableGroup()
magma: IsSimple(G);
gap: IsSimpleGroup(G);
sage_gap: 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_gap: G.CharacterDegrees()
Minimal presentations
Constructions
Groups of Lie type : $\SOPlus(6,2)$, $\GOPlus(6,2)$, $\PSOPlus(6,2)$, $\PGOPlus(6,2)$, $\CSOPlus(6,2)$, $\COPlus(6,2)$ Permutation group :
$\langle(1,2,3,4,5,6,7,8), (1,2)\rangle$
magma: G := PermutationGroup< 8 | (1,2,3,4,5,6,7,8), (1,2) >;
gap: G := Group( (1,2,3,4,5,6,7,8), (1,2) );
sage: G = PermutationGroup(['(1,2,3,4,5,6,7,8)', '(1,2)'])
Transitive group :8T50 16T1838 28T502 30T1153 all 5
Direct product :not isomorphic to a non-trivial direct product
Semidirect product :$A_8$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product :not isomorphic to a non-trivial transitive wreath product
Aut. group :$\Aut(A_8)$ $\Aut(C_2.A_8)$ $\Aut(C_2\times A_8)$
Elements of the group are displayed as permutations of degree 8.
Homology
Abelianization : $C_{2} $
magma: quo< G | CommutatorSubgroup(G) >;
gap: FactorGroup(G, DerivedSubgroup(G));
sage: G.quotient(G.commutator())
Schur multiplier :$C_{2}$
gap: AbelianInvariantsMultiplier(G);
sage: G.homology(2)
sage_gap: G.AbelianInvariantsMultiplier()
Commutator length :$1$
gap: CommutatorLength(G);
sage_gap: G.CommutatorLength()
Subgroups
magma: Subgroups(G);
gap: AllSubgroups(G);
sage: G.subgroups()
sage_gap: G.AllSubgroups()
There are 151221 subgroups in 296 conjugacy classes , 3 normal , and all normal subgroups are characteristic.
Characteristic subgroups are shown in this color .
Special subgroups
Center :$Z \simeq$ $C_1$ $G/Z \simeq$ $S_8$
magma: Center(G);
gap: Center(G);
sage: G.center()
sage_gap: G.Center()
Commutator :$G' \simeq$ $A_8$ $G/G' \simeq$ $C_2$
magma: CommutatorSubgroup(G);
gap: DerivedSubgroup(G);
sage: G.commutator()
sage_gap: G.DerivedSubgroup()
Frattini :$\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $S_8$
magma: FrattiniSubgroup(G);
gap: FrattiniSubgroup(G);
sage: G.frattini_subgroup()
sage_gap: G.FrattiniSubgroup()
Fitting :$\operatorname{Fit} \simeq$ $C_1$ $G/\operatorname{Fit} \simeq$ $S_8$
magma: FittingSubgroup(G);
gap: FittingSubgroup(G);
sage: G.fitting_subgroup()
sage_gap: G.FittingSubgroup()
Radical :$R \simeq$ $C_1$ $G/R \simeq$ $S_8$
magma: Radical(G);
gap: SolvableRadical(G);
sage_gap: G.SolvableRadical()
Socle :$\operatorname{soc} \simeq$ $A_8$ $G/\operatorname{soc} \simeq$ $C_2$
magma: Socle(G);
gap: Socle(G);
sage: G.socle()
sage_gap: G.Socle()
2-Sylow subgroup :
$P_{ 2 } \simeq$ $C_2\wr D_4$
3-Sylow subgroup :
$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup :
$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup :
$P_{ 7 } \simeq$ $C_7$
Hi
diagram
profile
all subgroups
normal subgroups
up to conjugacy
up to automorphism
Normal subgroups
Normal subgroups up to automorphism
Classes of subgroups up to conjugation
Order 40320: $S_8$ Order 20160: $A_8$ Order 5040: $S_7$ Order 2520: $A_7$ Order 1440: $C_2\times S_6$ Order 1344: $C_2^3:\GL(3,2)$ Order 1152: $S_4\wr C_2$ Order 720: $S_6$ x 2, $S_3\times S_5$ , $C_2\times A_6$ Order 576: $\POPlus(4,3)$ , $S_4^2$ , $A_4^2:C_4$ Order 384: $C_2\wr S_4$ Order 360: $S_3\times A_5$ , $C_3:S_5$ , $C_3\times S_5$ , $A_6$ Order 336: $\PGL(2,7)$ Order 288: $\PSOPlus(4,3)$ , $A_4\wr C_2$ , $A_4\times S_4$ Order 240: $C_2\times S_5$ x 2 Order 192: $C_2^3:S_4$ , $C_2\wr A_4$ , $Q_8:S_4$ , $C_2^3:S_4$ , $D_4\times S_4$ Order 180: $\GL(2,4)$ Order 168: $\PSL(2,7)$ x 2, $F_8:C_3$ Order 144: $S_3^2:C_2^2$ , $A_4^2$ , $S_3\times S_4$ Order 128: $C_2\wr D_4$ Order 120: $S_5$ x 4, $C_2\times A_5$ x 2, $S_3\times F_5$ Order 96: $C_2^2\times S_4$ x 3, $C_2^2:S_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:A_4$ , $Q_8:A_4$ , $D_4\times A_4$ , $C_4:S_4$ , $C_4\times S_4$ Order 72: $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_2\times C_3^2:C_4$ , $S_3\times A_4$ , $C_3:S_4$ , $C_3\times S_4$ Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$ , $D_4^2$ , $D_4:D_4$ Order 60: $A_5$ x 2, $S_3\times D_5$ , $C_{15}:C_4$ , $C_3\times F_5$ Order 56: $F_8$ Order 48: $C_2\times S_4$ x 12, $C_2^2\times A_4$ x 3, $C_2^2:A_4$ , $S_3\times D_4$ , $C_4\times A_4$ , $A_4:C_4$ , $\GL(2,3)$ Order 42: $F_7$ Order 40: $C_2\times F_5$ Order 36: $S_3^2$ x 6, $C_3^2:C_4$ x 2, $C_6\times S_3$ x 2, $C_6:S_3$ , $C_3\times A_4$ Order 32: $C_2^2\wr C_2$ x 4, $C_2^3:C_4$ x 3, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $\OD_{16}:C_2$ , $D_4:C_2^2$ , $D_8:C_2$ , $C_4:D_4$ , $C_4\times D_4$ , $C_4\wr C_2$ Order 30: $D_{15}$ , $C_3\times D_5$ , $C_5\times S_3$ Order 24: $S_4$ x 10, $C_2\times A_4$ x 6, $C_2\times D_6$ x 3, $C_3:D_4$ x 2, $D_{12}$ , $C_4\times S_3$ , $\SL(2,3)$ , $C_3\times D_4$ Order 21: $C_7:C_3$ Order 20: $F_5$ x 2, $D_{10}$ Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$ Order 16: $C_2\times D_4$ x 14, $C_2^2:C_4$ x 6, $C_2^4$ x 3, $C_2^2\times C_4$ x 2, $\SD_{16}$ , $D_8$ , $\OD_{16}$ , $C_4:C_4$ , $C_4^2$ , $D_4:C_2$ Order 15: $C_{15}$ Order 14: $D_7$ Order 12: $D_6$ x 11, $A_4$ x 4, $C_2\times C_6$ x 3, $C_{12}$ , $C_3:C_4$ Order 10: $D_5$ x 2, $C_{10}$ Order 9: $C_3^2$ Order 8: $D_4$ x 14, $C_2^3$ x 10, $C_2\times C_4$ x 7, $Q_8$ , $C_8$ Order 7: $C_7$ Order 6: $S_3$ x 7, $C_6$ x 5 Order 5: $C_5$ Order 4: $C_2^2$ x 11, $C_4$ x 4 Order 3: $C_3$ x 2 Order 2: $C_2$ x 4 Order 1: $C_1$
Classes of subgroups up to automorphism
Order 40320: $S_8$ Order 20160: $A_8$ Order 5040: $S_7$ Order 2520: $A_7$ Order 1440: $C_2\times S_6$ Order 1344: $C_2^3:\GL(3,2)$ Order 1152: $S_4\wr C_2$ Order 720: $S_6$ x 2, $S_3\times S_5$ , $C_2\times A_6$ Order 576: $\POPlus(4,3)$ , $S_4^2$ , $A_4^2:C_4$ Order 384: $C_2\wr S_4$ Order 360: $S_3\times A_5$ , $C_3:S_5$ , $C_3\times S_5$ , $A_6$ Order 336: $\PGL(2,7)$ Order 288: $\PSOPlus(4,3)$ , $A_4\wr C_2$ , $A_4\times S_4$ Order 240: $C_2\times S_5$ x 2 Order 192: $C_2^3:S_4$ , $C_2\wr A_4$ , $Q_8:S_4$ , $C_2^3:S_4$ , $D_4\times S_4$ Order 180: $\GL(2,4)$ Order 168: $\PSL(2,7)$ x 2, $F_8:C_3$ Order 144: $S_3^2:C_2^2$ , $A_4^2$ , $S_3\times S_4$ Order 128: $C_2\wr D_4$ Order 120: $S_5$ x 4, $C_2\times A_5$ x 2, $S_3\times F_5$ Order 96: $C_2^2\times S_4$ x 3, $C_2^2:S_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:A_4$ , $Q_8:A_4$ , $D_4\times A_4$ , $C_4:S_4$ , $C_4\times S_4$ Order 72: $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 2, $C_2\times C_3^2:C_4$ , $S_3\times A_4$ , $C_3:S_4$ , $C_3\times S_4$ Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$ , $D_4^2$ , $D_4:D_4$ Order 60: $A_5$ x 2, $S_3\times D_5$ , $C_{15}:C_4$ , $C_3\times F_5$ Order 56: $F_8$ Order 48: $C_2\times S_4$ x 12, $C_2^2\times A_4$ x 3, $C_2^2:A_4$ , $S_3\times D_4$ , $C_4\times A_4$ , $A_4:C_4$ , $\GL(2,3)$ Order 42: $F_7$ Order 40: $C_2\times F_5$ Order 36: $S_3^2$ x 6, $C_3^2:C_4$ x 2, $C_6\times S_3$ x 2, $C_6:S_3$ , $C_3\times A_4$ Order 32: $C_2^2\wr C_2$ x 4, $C_2^3:C_4$ x 3, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $\OD_{16}:C_2$ , $D_4:C_2^2$ , $D_8:C_2$ , $C_4:D_4$ , $C_4\times D_4$ , $C_4\wr C_2$ Order 30: $D_{15}$ , $C_3\times D_5$ , $C_5\times S_3$ Order 24: $S_4$ x 10, $C_2\times A_4$ x 6, $C_2\times D_6$ x 3, $C_3:D_4$ x 2, $D_{12}$ , $C_4\times S_3$ , $\SL(2,3)$ , $C_3\times D_4$ Order 21: $C_7:C_3$ Order 20: $F_5$ x 2, $D_{10}$ Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$ Order 16: $C_2\times D_4$ x 14, $C_2^2:C_4$ x 6, $C_2^4$ x 3, $C_2^2\times C_4$ x 2, $\SD_{16}$ , $D_8$ , $\OD_{16}$ , $C_4:C_4$ , $C_4^2$ , $D_4:C_2$ Order 15: $C_{15}$ Order 14: $D_7$ Order 12: $D_6$ x 11, $A_4$ x 4, $C_2\times C_6$ x 3, $C_{12}$ , $C_3:C_4$ Order 10: $D_5$ x 2, $C_{10}$ Order 9: $C_3^2$ Order 8: $D_4$ x 14, $C_2^3$ x 10, $C_2\times C_4$ x 7, $Q_8$ , $C_8$ Order 7: $C_7$ Order 6: $S_3$ x 7, $C_6$ x 5 Order 5: $C_5$ Order 4: $C_2^2$ x 11, $C_4$ x 4 Order 3: $C_3$ x 2 Order 2: $C_2$ x 4 Order 1: $C_1$
Normal subgroups (quotient in parentheses)
Order 40320: $S_8$ ($C_1$) Order 20160: $A_8$ ($C_2$) Order 1: $C_1$ ($S_8$)
Normal subgroups up to automorphism (quotient in parentheses)
Order 40320: $S_8$ ($C_1$) Order 20160: $A_8$ ($C_2$) Order 1: $C_1$ ($S_8$)
Series
Derived series
$S_8$
$\rhd$
$A_8$
magma: DerivedSeries(G);
gap: DerivedSeriesOfGroup(G);
sage: G.derived_series()
sage_gap: G.DerivedSeriesOfGroup()
Chief series
$S_8$
$\rhd$
$A_8$
$\rhd$
$C_1$
magma: ChiefSeries(G);
gap: ChiefSeries(G);
sage_gap: G.ChiefSeries()
Lower central series
$S_8$
$\rhd$
$A_8$
magma: LowerCentralSeries(G);
gap: LowerCentralSeriesOfGroup(G);
sage: G.lower_central_series()
sage_gap: G.LowerCentralSeriesOfGroup()
Upper central series
$C_1$
magma: UpperCentralSeries(G);
gap: UpperCentralSeriesOfGroup(G);
sage: G.upper_central_series()
sage_gap: G.UpperCentralSeriesOfGroup()
Supergroups
This group is a maximal subgroup of 15 larger groups in the database.
This group is a maximal quotient of 22 larger groups in the database.
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_gap: G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
Every character has rational values, so the complex character table is the same as the rational character table below.
See the $22 \times 22$ rational character table .
Alternatively, you may search for characters of this group with desired properties.
