C₇₁×#7207 #31828079

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7207

Subgroups

	subgroup	G/H
center	C₈×C₇₁	#99
commutator	C₂³	C₄×C₈×C₇₁
Frattini, cosocle	C₂×#131	C₂²×C₇₁
socle	C₂×C₇₁	#1282
prime radical	C₂×#41×C₇₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₁ ◁ C₂×C₈×C₇₁ ◁ C₂×#131×C₇₁ ◁ G

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 2 generators on 135 points:

(1,2,5,11,7,14,27,38,8,13,6,12,26,37,28,39)(3,15,17,40,19,43,48,62,20,42,18,41,47,61,49,63)(4,21,22,36,23,50,51,60,24,10,25,33,52,34,53,58)(9,29,30,54,32,55,56,16,31,44,35,45,57,64,59,46)(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);(1,3,6,18,7,19,28,49,8,20,5,17,26,47,27,48)(2,9,12,30,14,32,39,56,13,31,11,35,37,57,38,59)(4,16,22,44,23,45,51,64,24,46,25,29,52,54,53,55)(10,15,33,41,34,43,58,63,21,42,36,40,50,61,60,62)

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, 195);