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C₃×#20851 #10639458

order	768
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂⁴×C₃×C₄ ◁ G

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9,12,16,13,15,11,20)(10,17,18,14,19,23,21,22);(8,10,12,18,13,19,11,21)(9,14,16,23,15,22,20,17);(5,7);(9,15)(14,22)(16,20)(17,23)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270305, #72270311, D₃×#20851, …
maximal subgroup of	C₃×#395409, #72270305, #72270311, …
maximal normal subgroup of	C₃×#395409, #72270305, #72270311, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13839);