C₇₁×#13227 #31834099

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₁×#13227

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₁
prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×C₄×Q₈×C₇₁ ◁ G

472 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 91
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 91 points:

(2,4)(5,6,9,12,10,13,20,18)(7,14,16,15,17,8,11,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(5,7,15,6,10,17,19,13)(8,18,9,11,14,12,20,16);(1,2,3,4)

Conjugacy classes & character table

2627 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6215);