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C₂²×#213 #5792

order	192
isoclinism	isoclinic class of #213
automorphism	Aut(G) order 36864
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#213

Splittings

Fitting	C₂²×C₃×D₄ ⋊ C₂
Sylow	C₃ ⋊ C₂²×D₈

Subgroups

	subgroup	G/H
center	C₂³	#67
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×D₄	C₂
socle	C₂³×C₃	D₄
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂²×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69647, #69648, #69650, …
maximal normal subgroup of	#69647, #69648, #69650, …
maximal quotient of	C₂²×#807, C₂²×#809, C₂×#4967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1351);