C₂×#14554264 #72244200

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

Aliases

direct product

C₂×#14554264

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×C₃⁵ ◁ G

6460 conjugacy classes of subgroups; only the 646 normal subgroups are shown.

normal subgroups only

This group has 6460 conjugacy classes of subgroups — too many to draw. Only its 646 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) = 38
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 38 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221889, C₂×#72221881, C₂×#72221909
maximal subgroup of	C₂×#72221849, C₂×#72221889, C₂×#72221881, …
maximal normal subgroup of	C₂×#72221849, C₂×#72221889, C₂×#72221881, …

GAP

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

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