C₇₁×#7110 #31827982

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₁×#7110

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#327
commutator	C₂×C₈	C₂×C₈×C₇₁
Frattini, cosocle	C₂×#97	C₂²×C₇₁
prime radical	C₂²×D₈×C₇₁	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#97×C₇₁ ◁ G

270 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 95
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 2 generators on 95 points:

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

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, 98);