Finite Group Databasebeta

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C₂×#622 #3791

order	144
isoclinism	isoclinic class of #157
automorphism	Aut(G) = D₄×AΓL(1,𝔽₉) (order 1152) · Inn(G) = #157 · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

Splittings

C₃² ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#157
commutator, hypocenter	C₃²	C₂×C₈
Frattini	C₂	C₂×#157
Fitting, socle, prime radical	C₂²×C₃²	C₄
cosocle	C₃⋊Dic₃	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11771727, #11771744, #11771761, …
maximal subgroup of	#63739, #63740, #63742, …
maximal normal subgroup of	#63739, #63740, #63742, …
maximal quotient of	C₂×#3708, #63773, C₄×#622, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 134);