C₆₇×#7121 #30686372

order	17152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7121

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#102
commutator	C₂×C₈	C₂×C₈×C₆₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

222 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 94 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)(17,18,20,23,22,26,19,27,21,25,28,31,29,32,30,24)(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);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,30)(22,29)(24,26)(25,31)(27,32)

Conjugacy classes & character table

3886 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 109);