C₂×C₃²×#12067 #72255348

order	4608
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₈ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₂×C₃²×#12067

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	#442
commutator	C₈	C₂³×C₃²×C₈
Frattini, cosocle	C₄×C₈	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ C₂×C₃²×#322 ◁ G

3468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 2304

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₃⋊D₃×#12067
maximal subgroup of	C₂×C₃⋊D₃×#12067
maximal normal subgroup of	C₂×C₃⋊D₃×#12067

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

C2×C32×#12067 #72255348