C₆₇×#9497 #30688748

order	17152
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₇×#9497

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#386
commutator	C₂⁴	C₂²×C₄×C₆₇
Frattini, cosocle	C₂⁵	C₂³×C₆₇
prime radical	C₆₇×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂⁵×C₆₇ ◁ G

1184 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 2144

A minimal faithful permutation representation — 3 generators on 83 points:

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

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, 2485);