C₆₇×#10529 #30689780

order	17152
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#10529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂³×C₆₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 115
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

1876 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 3517);