C₇₁×#11653 #31832525

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#11653

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₂³×C₇₁	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

300 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 4641);