Finite Group Databasebeta

← Prev (#64326) Next (#64328) →

C₂²×#627 #64327

order	288
isoclinism	isoclinic class of #625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#627

Splittings

C₃² ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂²×D₃²
Fitting, socle, prime radical	C₂³×C₃²	C₂²
hypocenter	C₃²	C₂²×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³

339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591191, #10591195, #10591205, …
maximal normal subgroup of	#10591191, #10591195, #10591205, …
maximal quotient of	C₂×#63869, C₂×#63872, C₂×C₄×#627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 975);