C₂×C₃×#10624930 #72244184

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁵ · Schur multiplier (H₂) = C₃¹⁰

Aliases

direct product

C₂×C₃×#10624930

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃⁵
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×H(3,𝔽₃) ◁ G

14380 conjugacy classes of subgroups; only the 5704 normal subgroups are shown.

normal subgroups only

This group has 14380 conjugacy classes of subgroups — too many to draw. Only its 5704 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) = 32
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221887, C₂×#72221880, C₂×C₃×#13091647
maximal subgroup of	C₂×#72221849, C₂×#72221887, C₂×#72221880, …
maximal normal subgroup of	C₂×#72221849, C₂×#72221887, C₂×#72221880, …

GAP

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

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