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C₄×D₂₄ #4692

order	192
isoclinism	isoclinic class of #204
automorphism	Aut(G) = C₂²×D₃×#3003 (order 3072) · Inn(G) = D₁₂ · Out(G) = C₂⁴×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₂₄

Splittings

Fitting	C₃×C₄×C₈ ⋊ C₂
Sylow	C₃ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂×C₄	D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄×C₈	D₃
hypocenter	C₃	C₄×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2,3)(8,9,13,15,11,14,10,12);(2,3)(9,12)(10,13)(14,15);(4,5,6,7)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67465, #67471, #67504, …
maximal normal subgroup of	#67465, #67471, #67504, …
maximal quotient of	C₈×D₂₄, #68052, #68078, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 251);