C₇₁×#8798 #31829670

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

Aliases

direct product

C₇₁×#8798

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#382
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂³×C₇₁	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 119
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 119 points:

(1,2,6,10,7,11,5,15)(3,12,13,16,14,9,4,8)(17,18,22,28,23,26,21,27)(19,29,31,43,32,24,37,40)(20,33,35,42,36,44,38,25)(30,45,47,41,34,48,46,39)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3,5,4,7,14,6,13)(2,8,10,12,11,16,15,9)(17,19,21,31,23,32,22,37)(18,24,27,40,26,29,28,43)(20,34,35,47,36,30,38,46)(25,41,33,45,42,39,44,48);(1,4,5,14,7,13,6,3)(2,9,10,8,11,12,15,16)(17,20)(18,25,26,42)(19,30)(21,35)(22,38)(23,36)(24,39,29,41)(27,33,28,44)(31,46)(32,34)(37,47)(40,48,43,45)

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, 1786);