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C₂×C₂²⋊(C₂²⋊C₉) #64190

order	288
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×C₂²⋊(C₂²⋊C₉)

Splittings

derived	C₂⁴ ⋊ C₂×C₉
Sylow	C₂⁵ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₂×C₉
Frattini	C₃	C₂×C₂²⋊A₄
Fitting, socle, prime radical	C₂⁵×C₃	C₃
cosocle	C₂⁴×C₃	C₂×C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₂×C₃

282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 36

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

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11919944, #11919946, #11920036, …
maximal subgroup of	#379⋊C₉, C₂×#120⋊C₉, #10593111, …
maximal normal subgroup of	#379⋊C₉, C₂×#120⋊C₉, #10593111, …
maximal quotient of	C₄×C₂²⋊(C₂²⋊C₉), C₂×C₂²⋊(Q₈⋊C₉), #10593123, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 838);