C₇₁×#13561 #31834433

order	18176
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₁×#13561

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₁	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₇₁
Frattini, cosocle	#127	C₂³×C₇₁
prime radical	C₂×#453×C₇₁	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂³×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×#127×C₇₁ ◁ G

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

normal subgroups only

This group has 550 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)(3,10,12,6)(4,8,14,7)(5,9)(11,16,13,15)(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)(2,6)(4,13)(5,11)(7,10)(8,16)(9,15)(12,14);(3,11)(6,15)(10,16)(12,13)

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, 6549);