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C₂×Q₈×A₄ #5940

order	192
automorphism	Aut(G) = S₄×#1020⋊C₂ (order 4608) · Inn(G) = C₂²×A₄ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

C₂×Q₈×A₄

Splittings

C₂³×Q₈ ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×A₄
commutator, cosocle	C₂³	C₂³×C₃
Frattini	C₂	C₂³×A₄
Fitting	C₂³×Q₈	C₃
socle	C₂⁴	C₂²×C₃
prime radical	C₂²×A₄	C₂²
hypercenter	C₂×Q₈	A₄
hypocenter	C₂²	C₂×C₃×Q₈
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	SL(2,𝔽₃)×GL(2,ℤ/4)
maximal subgroup of	#73146, #73147, A₄×#101, …
maximal normal subgroup of	#73146, #73147, A₄×#101, …
maximal quotient of	C₂×A₄×C₄⋊C₄, C₄×Q₈×A₄, A₄×C₂²⋊Q₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1499);