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S4×3−1+4 #72259502

order5832
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C2×C35

Aliases

direct productS4×3−1+4

Splittings

FittingC22×3−1+4 ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC3C34×S4
commutator, cosocleC3×A4C2×C34
FittingC22×3−1+4D3
socleC22×C3C34×D3
prime radicalC3×S4×H(3,𝔽3)C3
hypercenter3−1+4S4
hypocenterA4C2×3−1+4
LCSG ▷ C3×A4 ▷ A4
UCSC1 ◁ C3 ◁ 3−1+4

4864 conjugacy classes of subgroups; only the 852 normal subgroups are shown.

normal subgroups only

This group has 4864 conjugacy classes of subgroups — too many to draw. Only its 852 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 4 generators on 31 points:

(1,4,3,2)(7,15,14)(10,20,22)(13,24,23)(17,26,29)(19,27,28)(25,30,31);(2,3,4)(6,11,12)(7,14,15)(9,18,21)(10,20,22)(13,23,24)(25,30,31);(5,15,24,8,14,13,16,7,23)(6,20,27,12,22,19,11,10,28)(9,26,30,18,29,25,21,17,31);(5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,19,25)(14,22,29)(15,20,26)(23,28,31)(24,27,30)

Conjugacy classes & character table

415 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofS4×#10624749
maximal normal subgroup ofS4×#10624749

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,4,3,2)(7,15,14)(10,20,22)(13,24,23)(17,26,29)(19,27,28)(25,30,31), (2,3,4)(6,11,12)(7,14,15)(9,18,21)(10,20,22)(13,23,24)(25,30,31), (5,15,24,8,14,13,16,7,23)(6,20,27,12,22,19,11,10,28)(9,26,30,18,29,25,21,17,31), (5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,19,25)(14,22,29)(15,20,26)(23,28,31)(24,27,30));