C₂×C₃×#13091120 #72269360

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×C₃×#13091120

Splittings

Fitting	C₂×C₃×#10624749 ⋊ C₂
derived	#10624749 ⋊ C₂²×C₃
Sylow	C₃×#10624749 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#13091120
commutator, cosocle, hypocenter	#10624749	C₂²×C₃
Frattini	C₃³	C₂×C₃×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃×#10624749	C₂
socle	C₂×C₃²	#92338
LCS	G ▷ #10624749
UCS	C₁ ◁ C₂×C₃

5042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 32 points:

(3,4,5)(7,10)(8,17,16,13,14,28)(9,18)(11,31,15,22,23,29)(12,20)(19,25,32,27,24,30)(21,26);(1,2)(6,16,13,18,14,17,9,8,28)(7,15,27,26,23,25,12,11,30)(10,32,22,20,24,31,21,19,29);(8,16,14)(13,17,28)(19,24,32)(25,27,30);(6,7,10)(8,15,19)(9,12,21)(11,24,14)(13,27,29)(16,23,32)(17,25,22)(18,26,20)(28,30,31)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#13091120
maximal quotient of	AGL(1,𝔽₇)×#13091120

GAP

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

gap> G := Group((3,4,5)(7,10)(8,17,16,13,14,28)(9,18)(11,31,15,22,23,29)(12,20)(19,25,32,27,24,30)(21,26), (1,2)(6,16,13,18,14,17,9,8,28)(7,15,27,26,23,25,12,11,30)(10,32,22,20,24,31,21,19,29), (8,16,14)(13,17,28)(19,24,32)(25,27,30), (6,7,10)(8,15,19)(9,12,21)(11,24,14)(13,27,29)(16,23,32)(17,25,22)(18,26,20)(28,30,31));

C2×C3×#13091120 #72269360