C₂×C₃×UT(4,𝔽₃) #72244197

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁸

Aliases

direct product

C₂×C₃×UT(4,𝔽₃)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴ ◁ G

5632 conjugacy classes of subgroups; only the 628 normal subgroups are shown.

normal subgroups only

This group has 5632 conjugacy classes of subgroups — too many to draw. Only its 628 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 — 4 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×C₃×#13091108, C₂×C₃×#13091105, C₂×C₃×#13091107
maximal subgroup of	C₂×#72221882, C₂×#72221849, C₂×C₃×#13091108, …
maximal normal subgroup of	C₂×#72221882, C₂×#72221849, C₂×C₃×#13091108, …

GAP

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

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