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C₄×#622 #63775

order	288
isoclinism	isoclinic class of #157
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₃×C₄

Aliases

Splittings

C₃² ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#157
commutator, hypocenter	C₃²	C₄×C₈
Frattini	C₂²	C₂×#157
Fitting	C₂×C₃²×C₄	C₄
socle, prime radical	C₂²×C₃²	C₂×C₄
cosocle	C₂×C₃⋊Dic₃	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590029, #10590032, #10590041, …
maximal normal subgroup of	#10590029, #10590032, #10590041, …
maximal quotient of	C₄×#3708, #10590148, C₈×#622, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 423);