C₂×#14554265 #72244199

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

Aliases

direct product

C₂×#14554265

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃⁴
prime radical	C₂×C₃×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁵ ◁ G

4948 conjugacy classes of subgroups; only the 646 normal subgroups are shown.

normal subgroups only

This group has 4948 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)(14,20,30)(17,26,21)(22,33,31)(24,27,36)(28,32,34)(35,38,37);(3,4,6)(5,8,10)(7,11,9)(12,13,16)(14,21,24)(15,19,25)(17,27,20)(18,29,23)(22,34,35)(26,36,30)(28,38,33)(31,32,37);(4,7,8)(6,10,11);(12,14,22,15,20,31,23,30,33)(13,17,28,19,26,34,18,21,32)(16,24,35,25,27,37,29,36,38)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221908, C₂×#72221896, C₂×#72221943
maximal subgroup of	C₂×#72221882, C₂×#72221908, C₂×#72221896, …
maximal normal subgroup of	C₂×#72221882, C₂×#72221908, C₂×#72221896, …

GAP

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

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