C₇₁×#12885 #31833757

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

Aliases

direct product

C₇₁×#12885

Subgroups

	subgroup	G/H
center	C₈×C₇₁	#124
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
socle	C₂×C₇₁	#1832
prime radical	C₂³×C₇₁	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₁ ◁ C₂²×C₈×C₇₁ ◁ G

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 135
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2,6,12,7,13,30,45,8,14,31,46,5,11,27,42)(3,15,19,47,17,36,51,43,20,34,28,59,18,9,32,35)(4,21,25,54,24,41,57,44,26,39,29,60,23,10,37,40)(16,48,49,62,22,55,56,64,50,33,53,61,52,38,58,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3,5,18,8,20,7,17)(2,9,11,34,14,36,13,15)(4,22,24,50,26,52,23,16)(6,28,27,51,31,19,30,32)(10,38,21,48,41,55,39,33)(12,43,42,47,46,35,45,59)(25,58,57,49,29,56,37,53)(40,64,54,61,44,63,60,62);(1,4,5,23,8,26,7,24)(2,10,11,39,14,41,13,21)(3,16,17,22,20,50,18,52)(6,29,27,57,31,25,30,37)(9,33,15,38,36,48,34,55)(12,44,42,54,46,40,45,60)(19,53,51,58,28,49,32,56)(35,62,47,64,43,61,59,63)

Conjugacy classes & character table

3692 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5873);