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C₂×D₇₂ #63466

order	288
isoclinism	isoclinic class of Dic₃₆
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₇₂

Splittings

Fitting	C₂×C₈×C₉ ⋊ C₂
Sylow	C₉ ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂²	D₃₆
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₂×C₈×C₉	C₂
socle	C₂²×C₃	D₁₂
hypercenter	C₂×C₈	D₉
hypocenter	C₉	C₂×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10588168, #10588172, #10588186, …
maximal normal subgroup of	#10588168, #10588172, #10588186, …
maximal quotient of	#10588362, C₄×D₇₂, #10588376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 114);