C₆₇×#7790 #30687041

order	17152
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#7790

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₆₇×#2682	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

516 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 99 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28)(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);(2,5)(3,8)(6,7)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,28,30,31);(10,13)(11,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)

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, 778);