C₃²×#93877 #72245489

order	4608
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₁₆ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#93877

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#1368
commutator	C₁₆	C₂×C₃²×C₁₆
Frattini, cosocle	#1311	C₂²×C₃²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₈² ◁ C₃²×#1311 ◁ G

348 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 54
transitive degree	μ_t(G) = 4608

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,33,25,34,37,36)(19,27,28,41)(20,30)(21,32)(26,40,38,35,39,47,46,48)(29,44,43,42)(31,45)(49,50,51);(17,18,19,23,20,26,27,35,21,24,29,36,30,38,42,40,22,25,28,34,31,39,41,48,32,37,43,33,45,46,44,47)(52,53,54)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#93877, #72269796
maximal subgroup of	C₃⋊D₃×#93877, C₃²×#72224639, #72269796
maximal normal subgroup of	C₃⋊D₃×#93877, C₃²×#72224639, #72269796

GAP

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

gap> G := Group((1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,33,25,34,37,36)(19,27,28,41)(20,30)(21,32)(26,40,38,35,39,47,46,48)(29,44,43,42)(31,45)(49,50,51), (17,18,19,23,20,26,27,35,21,24,29,36,30,38,42,40,22,25,28,34,31,39,41,48,32,37,43,33,45,46,44,47)(52,53,54));