C₇₁×#7487 #31828359

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7487

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄²×C₇₁
Frattini, cosocle	C₂×#124	C₂²×C₇₁
prime radical	C₂²×D₄×C₇₁	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#124×C₇₁ ◁ G

298 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 91
transitive degree	μ_t(G) = 2272

A minimal faithful permutation representation — 2 generators on 91 points:

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

Conjugacy classes & character table

2414 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 475);