C₆₇×#9991 #30689242

order	17152
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₇×#9991

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 107
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 3 generators on 107 points:

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

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 2979);