C₆₇×#33304 #30712555

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

direct product

C₆₇×#33304

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

1762 conjugacy classes of subgroups; only the 414 normal subgroups are shown.

normal subgroups only

This group has 1762 conjugacy classes of subgroups — too many to draw. Only its 414 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) = 83
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 4 generators on 83 points:

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

Conjugacy classes & character table

3484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 26292);