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C₂×#953 #5840

order	192
automorphism	Aut(G) order 73728
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

Splittings

Fitting	C₂⁵×C₃ ⋊ C₂
Sylow	C₃ ⋊ C₂×#120

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂⁵×C₃	C₂
socle	C₂³×C₃	C₂³
hypercenter	C₂⁵	D₃
hypocenter	C₃	C₂×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 13 points:

(3,4,5)(6,7)(8,9);(1,2)(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	#69496, #69499, #69500, …
maximal normal subgroup of	#69496, #69499, #69500, …
maximal quotient of	C₂×#5210, C₂×#5213, C₄×#953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1399);