C₆₇×#13709 #30692960

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#13709

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
prime radical	#508×C₆₇	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

154 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 131 points:

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

Conjugacy classes & character table

2479 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6697);