#120×C₂₇⋊C₃ #72276249

order	2592
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#120×C₂₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₉	C₂³×C₃²
commutator, socle	C₂²×C₃	C₂³×C₃×C₉
prime radical	C₃²×#120	C₉
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

780 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

462 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#120×D₂₇⋊C₃
maximal subgroup of	#120×D₂₇⋊C₃
maximal normal subgroup of	#120×D₂₇⋊C₃

GAP

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

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