C₆₇×#8514 #30687765

order	17152
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₇×#8514

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#379
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₆₇×#3568	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

648 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Conjugacy classes & character table

2680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 1502);