C₂×C₃×#13091126 #72269379

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

Aliases

direct product

C₂×C₃×#13091126

Splittings

Fitting	C₂×C₃×#10624752 ⋊ C₂
derived	#10624752 ⋊ C₂²×C₃
Sylow	C₃×#10624752 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#13091126
commutator, cosocle, hypocenter	#10624752	C₂²×C₃
Frattini	C₃³	C₂×C₃×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃×#10624752	C₂
socle	C₂×C₃²	#92338
LCS	G ▷ #10624752
UCS	C₁ ◁ C₂×C₃

4748 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)(7,11)(8,26,17,15,16,18)(9,10)(12,21)(13,31,25,23,22,29)(14,19)(20,32,30,27,28,24);(1,2)(6,16,15,9,17,18,10,8,26)(7,22,27,12,25,24,14,13,32)(11,28,23,19,30,29,21,20,31);(7,12,14)(8,17,16)(11,19,21)(15,26,18);(6,7,11)(8,13,20)(9,12,19)(10,14,21)(15,27,23)(16,22,28)(17,25,30)(18,24,29)(26,32,31)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

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

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

C2×C3×#13091126 #72269379