#63551

order	288
isoclinism	isoclinic class of #63546
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	#3714
commutator, Fitting, cosocle	C₃²×C₈	C₂²
Frattini, hypercenter	C₈	D₃²
socle, prime radical	C₂×C₃²	D₈
hypocenter	C₃²	Q₃₂
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

57 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 38 points:

(1,3,12,28,5,14,18,20,6,15,23,32,17,29,4,13)(2,7,16,26,9,21,24,27,10,22,31,30,19,25,8,11)(34,35)(36,37,38);(1,2,6,10)(3,11,15,27)(4,16,18,31)(5,19,17,9)(7,20,22,13)(8,23,24,12)(14,30,29,26)(21,28,25,32)(33,34,35)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×Dic₂₄, #10590238, #10590239, …
maximal normal subgroup of	D₃×Dic₂₄, #10590238, #10590239, …
maximal quotient of	#10589118, #10589119, #10589139, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 199);