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D₃²×Q₈ #64317

order	288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

D₃²×Q₈

Splittings

Fitting	C₃²×Q₈ ⋊ C₂²
Sylow	C₃² ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₃²×Q₈	C₂²
prime radical	C₂×D₃²	C₂²
hypercenter	Q₈	D₃²
hypocenter	C₃²	C₂²×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10593471, #10593472, D₃²×SD₁₆, …
maximal normal subgroup of	#10593471, #10593472, D₃²×SD₁₆, …
maximal quotient of	#10594312, #10594313, #10594314, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 965);