C₆₇×#7128 #30686379

order	17152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7128

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₁₆×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
socle, prime radical	C₂³×C₆₇	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

236 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 17152

A minimal faithful permutation representation — 2 generators on 99 points:

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

Conjugacy classes & character table

5092 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 116);