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C₂×#875 #5476

order	192
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Splittings

Fitting	C₂×C₃×C₄² ⋊ C₂
Sylow	C₃ ⋊ C₂×#124

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
prime radical	C₂²×D₁₂	C₂
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	C₂×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 17 points:

(1,2)(3,4,5)(6,8,9,7)(10,12,13,16)(11,15,14,17);(6,7,9,8)(11,14)(15,17);(4,5)(7,8)(10,11)(12,15)(13,14)(16,17);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68630, #68675, #68677, …
maximal normal subgroup of	#68630, #68675, #68677, …
maximal quotient of	C₄×#875, #73308, #73313, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1035);