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C₂×#947 #5814

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

Aliases

Splittings

Fitting	C₂²×C₃×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#124

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴
Frattini	C₂²	C₂³×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂³×C₃	C₂³
prime radical	C₂²×D₁₂	C₂
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂×#124
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#70099, #70104, #70116, …
maximal normal subgroup of	#70099, #70104, #70116, …
maximal quotient of	C₂×#4977, C₂×#4990, C₂×#4988, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1373);