C₂×#29656⋊C₃ #72268344

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

direct product

C₂×#29656⋊C₃

Splittings

Fitting, Sylow

C₂×#29656 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120⋊C₃
commutator, cosocle, hypocenter	#543	C₂³×C₃
Frattini	C₂⁴	C₂³×A₄
Fitting	C₂×#29656	C₃
hypercenter	C₂⁴	#120⋊C₃
LCS	G ▷ #543
UCS	C₁ ◁ C₂³ ◁ C₂⁴

2086 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 2086 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2)(4,8,5)(7,10,9)(13,20,14)(17,24,18)(19,25,22)(21,26,23);(3,10,6,8)(4,5,7,9)(11,20)(12,25)(13,14)(15,26)(16,24)(17,22)(18,19)(21,23);(4,7)(5,9)(8,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,25)(24,26)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72268381

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(4,8,5)(7,10,9)(13,20,14)(17,24,18)(19,25,22)(21,26,23), (3,10,6,8)(4,5,7,9)(11,20)(12,25)(13,14)(15,26)(16,24)(17,22)(18,19)(21,23), (4,7)(5,9)(8,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,25)(24,26));