C₇₁×#7210 #31828082

order	18176
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₁×#7210

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	#95
commutator	C₂×C₄	C₄×C₈×C₇₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₁
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

206 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 9088

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

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

Conjugacy classes & character table

4544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 198);