C₆₇×#7085 #30686336

order	17152
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7085

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#327
commutator	C₂²×C₈	C₂×C₄×C₆₇
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₆₇
prime radical	C₂⁴×C₆₇	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₂²⋊C₈×C₆₇ ◁ G

196 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 99 points:

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

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, 73);