C₆₇×#12451 #30691702

order	17152
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#12451

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#466
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

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

normal subgroups only

This group has 224 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) = 91
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 3 generators on 91 points:

(1,2,4,7)(3,5,8,6)(10,14)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14);(2,5)(3,8)(6,7)

Conjugacy classes & character table

3283 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 5439);