C₆₇×#9201 #30688452

order	17152
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#9201

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁵×C₆₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

612 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 2144

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

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

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, 2189);