C₂×#72221885 #72244337

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃⁴

Aliases

direct product

C₂×#72221885

Splittings

Fitting	C₂×#14554854 ⋊ C₂
derived	C₃⁵ ⋊ C₂²×C₃²
Sylow	#14554854 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221885
commutator, cosocle, hypocenter	C₃⁵	C₂²×C₃²
Frattini	C₃³	C₂×C₃²×C₃⋊D₃
Fitting	C₂×#14554854	C₂
socle	C₂×C₃	#13091503
LCS	G ▷ C₃⁵
UCS	C₁ ◁ C₂

3328 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 3328 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 29
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 29 points:

(1,3,9)(2,6,15)(4,11,19,12,10,20)(5,13,21)(7,16,25,8,17,24)(14,22,26,18,23,27);(1,7,14,4,8,18,12,2,5)(3,6,13,10,17,23,11,16,22)(9,15,26,19,25,27,20,24,21)(28,29);(3,10,11)(6,16,17)(15,24,25)(21,26,27)

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72221898
maximal normal subgroup of	C₂×#72221898

GAP

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

gap> G := Group((1,3,9)(2,6,15)(4,11,19,12,10,20)(5,13,21)(7,16,25,8,17,24)(14,22,26,18,23,27), (1,7,14,4,8,18,12,2,5)(3,6,13,10,17,23,11,16,22)(9,15,26,19,25,27,20,24,21)(28,29), (3,10,11)(6,16,17)(15,24,25)(21,26,27));