C₇₁×#12280 #31833152

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#12280

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₇₁×#3191	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×C₂²⋊C₈×C₇₁ ◁ G

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,23)(19,22)(20,27)(24,32)(25,31)(26,29)(28,30)(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)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18)(19,24)(20,28)(21,25)(22,30)(23,27)(26,31)(29,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19,21,22)(18,24,25,30)(20,26,29,27)(23,28,31,32)

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, 5268);