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C₂×#874 #5475

order	192
automorphism	Aut(G) order 294912
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₂×C₃×C₄² ⋊ C₂
Sylow	C₃ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂×C₃×C₄²	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	C₂×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

498 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68629, #68674, #68676, …
maximal normal subgroup of	#68629, #68674, #68676, …
maximal quotient of	C₄×#874, #73311, #73313, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1034);