C₇₁×#13591 #31834463

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₁×#13591

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	C₂×#353
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇₁
prime radical	C₇₁×#3151	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ #532×C₇₁ ◁ G

450 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 87 points:

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

Conjugacy classes & character table

1988 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 6579);