C₆₇×#13420 #30692671

order	17152
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#13420

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₆₇
prime radical	C₂×#120×C₆₇	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₆₇ ◁ C₂²×D₄×C₆₇ ◁ G

600 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 87
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 87 points:

(2,3)(5,6,9,12)(7,13,15,20)(8,11,16,19)(10,17,18,14)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,2)(3,4)(8,16)(11,19);(1,3,4,2)(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,17)(19,20)

Conjugacy classes & character table

2077 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6408);