C₃²×#7389 #72255442

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

Aliases

direct product

C₃²×#7389

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#1333
commutator	C₁₆	C₂×C₃²×C₈
Frattini, cosocle	#346	C₂²×C₃²
prime radical	C₃²×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ C₃²×#346 ◁ G

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

normal subgroups only

This group has 354 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:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#7389, #72255466
maximal subgroup of	C₃⋊D₃×#7389, C₃²×AGL(1,ℤ/32), #72255466
maximal normal subgroup of	C₃⋊D₃×#7389, C₃²×AGL(1,ℤ/32), #72255466
maximal quotient of	C₃²×#93757

GAP

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

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