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C₂²×#216 #5809

order	192
isoclinism	isoclinic class of #213
automorphism	Aut(G) order 36864
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#216

Splittings

C₃ ⋊ C₂²×Q₁₆

Subgroups

	subgroup	G/H
center	C₂³	#67
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle, prime radical	C₂³×C₃	D₄
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂²×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(5,6,7)(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16);(6,7)(8,9,12,15)(10,16,18,23)(11,19,21,14)(13,20,22,17);(3,4);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69677, #69678, #69681, …
maximal normal subgroup of	#69677, #69678, #69681, …
maximal quotient of	C₂²×#807, C₂²×#810, C₂×#4970, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1368);