C₆₇×#10347 #30689598

order	17152
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#10347

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

500 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 4288

A minimal faithful permutation representation — 3 generators on 99 points:

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

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, 3335);