Finite Group Databasebeta

← Prev (#5738) Next (#5740) →

C₂²×#204 #5739

order	192
isoclinism	isoclinic class of #204
automorphism	Aut(G) order 36864
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#204

Splittings

Fitting	C₂²×C₃×C₈ ⋊ C₂
Sylow	C₃ ⋊ C₂²×SD₁₆

Subgroups

	subgroup	G/H
center	C₂³	D₁₂
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂³×C₃	D₄
prime radical	C₂²×D₁₂	C₂
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	C₂²×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

298 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)(8,9,10,12,11,13,14,15);(1,2)(3,4)(6,7)(9,12)(10,14)(13,15);(3,4);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68672, #68675, #68676, …
maximal normal subgroup of	#68672, #68675, #68676, …
maximal quotient of	C₂×#4680, C₂×#4694, C₂×#4693, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1298);