C₇₁×#12374 #31833246

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

Aliases

direct product

C₇₁×#12374

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₂×D₄×C₇₁	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ C₈×D₄×C₇₁ ◁ G

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

normal subgroups only

This group has 344 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) = 111
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,5,8)(3,6,4,7)(9,10,14,18)(11,19,22,38)(12,23,27,37)(13,28,25,17)(15,30,32,20)(16,33,35,26)(21,39,29,34)(24,36,40,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,6)(3,4)(7,8)(9,11,14,22)(10,15,18,32)(12,24,27,40)(13,29,25,21)(16,34,35,39)(17,36,28,31)(19,37,38,23)(20,33,30,26);(2,7)(3,4)(6,8)

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5362);