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C₂×#634 #3825

order	144
isoclinism	isoclinic class of #634
automorphism	Aut(G) = B(4,𝔽₂)×AGL(2,𝔽₃) (order 27648) · Inn(G) = C₂×C₃⋊D₃ · Out(G) = #11714905
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

C₃² ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂³
Frattini	C₂	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄	C₂
socle, prime radical	C₂²×C₃²	C₂²
hypercenter	C₂×C₄	C₃⋊D₃
hypocenter	C₃²	C₂×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11772775, #11772776, #11772791, …
maximal subgroup of	#63561, #63567, #63637, …
maximal normal subgroup of	#63561, #63567, #63637, …
maximal quotient of	C₄×#634, #64078, #64079, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 168);