C₆₇×#19710 #30698961

order	17152
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₆₇×#19710

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

772 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 99
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 4 generators on 99 points:

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

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, 12698);