C₇₁×#12774 #31833646

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₁×#12774

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₁
prime radical	C₂×H(3,𝔽₄)×C₇₁	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₇₁ ◁ C₂²×Q₈×C₇₁ ◁ G

932 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 91 points:

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

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, 5762);