C₂×C₃²×#1369 #72255494

order	2304
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₈ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₂×C₃²×#1369

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	#106
commutator	C₈	C₂²×C₃²×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃²
socle	C₂²×C₃²	#334
prime radical	C₂³×C₃²	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ C₂×C₃²×C₄×C₈ ◁ G

666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 40 points:

(3,4,6,12,8,14,23,30,10,15,24,13,9,26,7,25)(5,16,17,27,19,31,18,28,21,29,22,34,20,33,32,11)(35,36,37);(1,2)(38,39,40);(3,5,7,18,9,20,24,17,10,21,23,32,8,19,6,22)(4,11,13,16,14,27,25,31,15,28,12,29,26,34,30,33)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₃⋊D₃×#1369
maximal subgroup of	C₂×C₃⋊D₃×#1369
maximal normal subgroup of	C₂×C₃⋊D₃×#1369

GAP

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

gap> G := Group((3,4,6,12,8,14,23,30,10,15,24,13,9,26,7,25)(5,16,17,27,19,31,18,28,21,29,22,34,20,33,32,11)(35,36,37), (1,2)(38,39,40), (3,5,7,18,9,20,24,17,10,21,23,32,8,19,6,22)(4,11,13,16,14,27,25,31,15,28,12,29,26,34,30,33));

C2×C32×#1369 #72255494