C₄×#13091120 #72228530

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₄×#13091120

Splittings

Fitting	C₄×#10624749 ⋊ C₂
derived, Sylow	#10624749 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄	#13091120
commutator, hypocenter	#10624749	C₂×C₄
Frattini	C₂×C₃³	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₄×#10624749	C₂
socle	C₂×C₃	C₂×#92338
prime radical	C₂×#13091120	C₂
cosocle	C₂×#10624749	C₂²
LCS	G ▷ #10624749
UCS	C₁ ◁ C₄

1354 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

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

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#13091120, C₄×#72210336, AGL(1,𝔽₅)×#13091120, …
maximal normal subgroup of	D₄×#13091120, C₄×#72210336, #72259991
maximal quotient of	AGL(1,𝔽₅)×#13091120

GAP

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

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