C₆₇×#7391 #30686642

order	17152
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7391

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#1333
commutator	C₁₆	C₂×C₈×C₆₇
Frattini, cosocle	#346	C₂²×C₆₇
prime radical	C₂²×C₆₇	#361
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄×C₈×C₆₇ ◁ #346×C₆₇ ◁ G

116 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 131
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 2 generators on 131 points:

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

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 379);