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C₂×#940 #5803

order	192
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₂²×C₃×D₄ ⋊ C₂
Sylow	C₃ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂²×C₃×D₄	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4474, #70064, #70070, …
maximal normal subgroup of	C₂×#4474, #70064, #70070, …
maximal quotient of	C₂×#4957, C₂×#4949, #82115, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1362);