C₇₁×#12388 #31833260

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

Aliases

direct product

C₇₁×#12388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₇₁×#3403	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ #446×C₇₁ ◁ G

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

normal subgroups only

This group has 296 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) = 119
transitive degree	μ_t(G) = 9088

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

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,23,27)(19,28,31,46)(20,25,22,37)(21,33,36,44)(24,39,41,30)(26,32,45,35)(29,47,38,43)(34,40,48,42)(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);(4,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,40)(26,43)(27,41)(28,33)(30,35)(32,39)(36,48)(37,42)(44,46)(45,47);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,35,36,32)(24,40,41,42)(26,44,45,33)(28,43,46,47)(30,48,39,34)

Conjugacy classes & character table

2201 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5376);