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C₂²×#611 #63718

order	288
isoclinism	isoclinic class of Dic₁₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#611

Splittings

Fitting	C₂⁴×C₉ ⋊ C₂
Sylow	C₉ ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₉
commutator, cosocle	C₂×C₉	C₂⁴
Frattini	C₂×C₃	C₂³×D₃
Fitting	C₂⁴×C₉	C₂
socle	C₂³×C₃	C₂×D₃
hypercenter	C₂⁴	D₉
hypocenter	C₉	C₂²×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴

354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#63365, #10588636, #10588637, …
maximal normal subgroup of	C₂×#63365, #10588636, #10588637, …
maximal quotient of	C₂²×Dic₉⋊C₄, C₂×#63486, C₂²×(C₂×D₉)⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 366);