C₃²×#7457 #72245486

order	2304
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₈ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×#7457

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#1368
commutator	C₁₆	C₂×C₃²×C₈
Frattini, cosocle	#346	C₂²×C₃²
prime radical	C₂²×C₃²	#365
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ C₃²×#346 ◁ G

228 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 38
transitive degree	μ_t(G) = 288

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

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#7457, #72255461
maximal subgroup of	C₃⋊D₃×#7457, C₃²×AGL(1,ℤ/32), #72255461
maximal normal subgroup of	C₃⋊D₃×#7457, C₃²×AGL(1,ℤ/32), #72255461
maximal quotient of	C₃²×#93877

GAP

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

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