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C₂²×C₄×D₃ #999

order	96
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = D₃×#72209641 (order 9216) · Inn(G) = D₃ · Out(G) = #72209641
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×C₄×D₃

Splittings

Fitting	C₂²×C₃×C₄ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂³×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃
commutator, hypocenter	C₃	C₂³×C₄
Frattini	C₂	C₂³×D₃
Fitting	C₂²×C₃×C₄	C₂
socle	C₂³×C₃	C₂²
prime radical	C₂³×D₃	C₂
cosocle	C₂×C₃	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₄

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 11 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#95, #4664, #4665, …
maximal normal subgroup of	D₃×#95, #4664, #4665, …
maximal quotient of	C₂×C₄×Dic₆, C₂×C₄²×D₃, C₂×#872, …
automorphism group of	D₃×C₁₆, #798, C₅×C₃⋊C₈, …
outer automorphism group of	C₅×#131, C₃×#504, C₃×C₅×#41

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 206);