C₆₇×#10332 #30689583

order	17152
nilpotency class	3
isoclinism	isoclinic class of #10326
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#10332

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

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

normal subgroups only

This group has 600 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) = 99
transitive degree	μ_t(G) = 4288

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

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

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, 3320);