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Dic₃×Dic₆ #63842

order	288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₃² ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄	C₂²
socle, prime radical, cosocle	C₂²×C₃²	C₂³
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	C₄×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 18 points:

(5,7,8,12)(6,9,10,11)(13,14,15)(16,17,18);(5,6,8,10)(7,11,12,9)(17,18);(1,2,3,4)(14,15)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590475, #10590476, #10590477, …
maximal normal subgroup of	#10590475, #10590476, #10590477, …
maximal quotient of	C₃⋊C₈×Dic₆, #10590251, #10590258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 490);