C₇₁×#9410 #31830282

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₁×#9410

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
prime radical	C₂×#121×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

644 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 95
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 95 points:

(1,2,4,7,5,8,3,6)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(2,6,8,7)(9,11,13,18)(10,14)(12,17,20,23)(15,21)(16,22)(19,24);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)

Conjugacy classes & character table

3479 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 2398);