C₂×C₃²×#1333 #72255347

order	2304
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₈ · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product

C₂×C₃²×#1333

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	#102
commutator	C₈	C₂²×C₃²×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃²
prime radical	C₂²×C₃²×D₈	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ C₂×C₃²×C₄×C₈ ◁ G

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₃⋊D₃×#1333, C₂×#72222571
maximal subgroup of	C₂×C₃²×#12067, C₂×C₃⋊D₃×#1333, C₂×#72222571
maximal normal subgroup of	C₂×C₃²×#12067, C₂×C₃⋊D₃×#1333, C₂×#72222571

GAP

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

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

C2×C32×#1333 #72255347