C₆₇×#13721 #30692972

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₇×#13721

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
prime radical	C₂×D₁₆×C₆₇	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

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

normal subgroups only

This group has 180 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) = 139
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 3 generators on 139 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,25)(12,18)(13,35)(14,39)(15,41)(16,44)(17,24)(19,30)(20,26)(21,38)(22,36)(23,42)(27,49)(28,60)(29,50)(31,55)(32,54)(33,66)(34,48)(37,52)(40,51)(43,70)(45,64)(46,63)(47,65)(53,69)(56,61)(57,68)(58,67)(59,72)(62,71)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(2,5)(6,8)(9,11,26,32,15,28,57,65,42,61,40,59,69,64,38,34,17,30,14,27,43,62,37,33,16,29,58,63,36,31,13,12)(10,18,35,55,22,46,67,50,44,66,52,71,70,49,39,19,24,48,21,45,53,72,51,56,23,47,68,60,41,54,20,25);(1,3,4,7)(2,6,5,8)(9,12,13,31,36,63,58,29,16,33,37,62,43,27,14,30,17,34,38,64,69,59,40,61,42,65,57,28,15,32,26,11)(10,19,20,49,41,71,68,66,23,50,51,46,53,55,21,18,24,25,39,54,70,60,52,47,44,56,67,72,22,45,35,48)

Conjugacy classes & character table

4690 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6709);