C₃²×#7053 #72245341

order	2304
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₁₆ · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃²×#7053

Subgroups

	subgroup	G/H
center	C₃²×C₄²	C₄²
commutator	C₄	C₃²×C₄×C₁₆
Frattini, cosocle	C₈²	C₂²×C₃²
socle, prime radical	C₂²×C₃²	#322
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃²×C₄² ◁ G

366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 2304

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

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#7053, #72245416, #72269811
maximal subgroup of	C₃²×#93668, C₃⋊D₃×#7053, #72245416, …
maximal normal subgroup of	C₃²×#93668, C₃⋊D₃×#7053, #72245416, …
maximal quotient of	C₃²×#93668

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)(18,22,23,28)(19,25)(24,31,30,29)(26,32)(33,34,35), (17,18,19,24,20,22,26,29,21,23,25,30,27,28,32,31)(36,37,38));

C32×#7053 #72245341