C₇₁×#9973 #31830845

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₁×#9973

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₁	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂⁴×C₇₁	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

632 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 99
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 99 points:

(1,2)(3,4)(5,6,10,7)(8,11,9,12)(13,14,19,16)(15,22,25,28)(17,27,18,21)(20,24,26,23)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(6,11)(7,12)(8,9)(13,15,19,25)(14,20,16,26)(17,24,18,23)(21,28,27,22);(2,4)(5,7,10,6)(8,12,9,11)(13,16,19,14)(15,22,25,28)(17,27,18,21)(20,23,26,24)

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 2961);