C₆₇×#10518 #30689769

order	17152
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#10518

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₆₇×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

478 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 83 points:

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

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, 3506);