C₇₁×#7195 #31828067

order	18176
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#7195

Subgroups

	subgroup	G/H
center	C₄²×C₇₁	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₇₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₁
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

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

normal subgroups only

This group has 222 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) = 103
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 2 generators on 103 points:

(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)

Conjugacy classes & character table

6248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 183);