C₃×#72214532 #72243997

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

Aliases

direct product

C₃×#72214532

Splittings

Fitting, Sylow	C₃×UT(4,𝔽₃) ⋊ C₂²
derived	UT(4,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72214532
commutator, cosocle, hypocenter	UT(4,𝔽₃)	C₂²×C₃
Frattini	C₃³	C₃×#1113
Fitting	C₃×UT(4,𝔽₃)	C₂²
socle	C₃²	#11757749
LCS	G ▷ UT(4,𝔽₃)
UCS	C₁ ◁ C₃

2107 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 2107 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 30
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244007, #72244009, #72244010, …
maximal normal subgroup of	C₃×#72214626

GAP

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

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