C₆₇×#12394 #30691645

order	17152
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#12394

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₇
prime radical	C₂×D₄×C₆₇	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 8576

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

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

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, 5382);