#64113

order	288
isoclinism	isoclinic class of #3744
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×C₈ ⋊ C₂
Sylow	C₃² ⋊ #135

Subgroups

	subgroup	G/H
center	C₄	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂×C₃²	C₂×D₄
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	#135
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

(1,4,5,12,6,13,3,10)(2,8,7,11,9,15,14,16)(20,21,22);(1,3,6,5)(2,7,9,14)(4,10,13,12)(8,11,15,16)(17,18,19);(1,2)(3,7)(4,11)(5,14)(6,9)(8,12)(10,15)(13,16)(18,19)(21,22)

Polycyclic presentation:

2^5,3^2|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=g^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589122, #10589125, #10589354, …
maximal normal subgroup of	#10589122, #10589125, #10589354, …
maximal quotient of	#10592382, C₄×#3744, C₄×#3745, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 761);