C₆₇×#11559 #30690810

order	17152
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#11559

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₆₇
prime radical	C₂×#120×C₆₇	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

532 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 2144

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

(1,2,4,8,5,6,3,7)(9,10)(11,15)(12,19,20,14)(13,16)(17,23,24,21)(18,22)(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);(2,6)(3,4)(10,14)(15,21)(16,19)(22,23);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,16,22)(12,17,20,24)(14,21,19,23)

Conjugacy classes & character table

2479 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 4547);