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C₂³×Dic₃ #1011

order	96
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 129024
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×Dic₃

Splittings

C₃ ⋊ C₂³×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴	D₃
commutator, hypocenter	C₃	C₂³×C₄
Frattini	C₂	C₂³×D₃
Fitting, socle, prime radical	C₂⁴×C₃	C₂
cosocle	C₂×C₃	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 13 points:

(1,2)(3,4)(5,6)(8,9)(10,11,12,13);(5,6)(7,8,9);(3,4);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×C₂²⋊C₄, #4942, #4943, …
maximal normal subgroup of	Dic₃×C₂²⋊C₄, #4942, #4943, …
maximal quotient of	C₂³×C₃⋊C₈, C₂²×#208, C₂²×C₄×Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 218);