C₇₁×#11166 #31832038

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₁×#11166

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#381
commutator	C₂⁴	C₂²×C₄×C₇₁
Frattini, cosocle	C₂⁵	C₂³×C₇₁
prime radical	C₂³×D₄×C₇₁	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂⁵×C₇₁ ◁ G

1110 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1110 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 2272

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

(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(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);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(2,4)(5,8)(6,11,12,16)(7,14)(9,17)(10,18,13,19)(15,20)

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 4154);