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C₂²×D₂₄ #5740

order	192
isoclinism	isoclinic class of #204
automorphism	Aut(G) order 73728
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×D₂₄

Splittings

Fitting	C₂²×C₃×C₈ ⋊ C₂
Sylow	C₃ ⋊ C₂²×D₈

Subgroups

	subgroup	G/H
center	C₂³	D₁₂
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂³×C₃	D₄
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	C₂²×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68673, #68674, #68675, …
maximal normal subgroup of	#68673, #68674, #68675, …
maximal quotient of	C₂×#4682, C₂×C₄×D₂₄, C₂×#4694, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1299);