C₇₁×#9414 #31830286

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

Aliases

direct product

C₇₁×#9414

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

552 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 103
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 103 points:

(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,22,26,31)(20,27,29,23)(21,24)(25,30)(28,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,21,26)(18,22)(20,28,29,25)(23,32)(24,31)(27,30);(3,11)(7,15)(10,16)(12,13)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)

Conjugacy classes & character table

2627 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 2402);