C₆₇×#19073 #30698324

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₇×#19073

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

1252 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 4288

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

(2,6)(3,4)(7,8)(10,14)(11,12)(15,16)(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);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,2)(3,6)(4,7)(5,8)(11,12)(14,16);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,15,16)

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, 12061);