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D₃×C₄⋊Q₈ #5723

order	192
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

D₃×C₄⋊Q₈

Splittings

Fitting	C₃×C₄⋊Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×C₄⋊Q₈

Subgroups

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

290 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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#332, #68420, #68422, …
maximal normal subgroup of	D₃×#332, #68420, #68422, …
maximal quotient of	#74320, #74326, #74329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1282);