C₇₁×#7440 #31828312

order	18176
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇₁×#7440

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	#1353
commutator	C₂×M₁₆	C₂×C₄×C₇₁
Frattini, cosocle	#423	C₂²×C₇₁
prime radical	C₂³×C₇₁	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ #423×C₇₁ ◁ G

144 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 87
transitive degree	μ_t(G) = 1136

A minimal faithful permutation representation — 2 generators on 87 points:

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

Conjugacy classes & character table

1775 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 428);