C₇₁×#7516 #31828388

order	18176
nilpotency class	6
isoclinism	isoclinic class of #7515
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₁×#7516

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	#1402
commutator	#127	C₂×C₄×C₇₁
Frattini, cosocle	#493	C₂²×C₇₁
prime radical	#493×C₇₁	C₂²
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂²×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₄²×C₇₁ ◁ #493×C₇₁ ◁ G

162 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 30 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:

(1,2)(3,8,11,16)(4,12,13,15)(5,9)(6,10,7,14)(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);(1,3,5,11)(2,6,9,12)(4,10,14,13)(7,8,15,16)

Conjugacy classes & character table

1562 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 504);