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C₄×C₉⋊C₈ #63361

order	288
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

C₄×C₉⋊C₈

Splittings

C₉ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	D₉
commutator, hypocenter	C₉	C₄×C₈
Frattini	C₂×C₃×C₄	C₂×D₃
Fitting	C₄²×C₉	C₂
socle, prime radical	C₂²×C₃	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂²
LCS	G ▷ C₉
UCS	C₁ ◁ C₄²

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 21 points:

(2,4)(3,7)(5,6)(8,9)(14,15,16,18,17,19,20,21);(1,9,4,7,6,5,3,2,8)(10,11,12,13)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×C₉⋊C₈, #10588131, #10588132, …
maximal normal subgroup of	C₈×C₉⋊C₈, #10588131, #10588132, …
maximal quotient of	C₈×C₉⋊C₈, #10588132, C₄×C₉⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 9);