C₆₇×#32900 #30712151

order	17152
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₆₇×#32900

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂²×H(5,𝔽₂)×C₆₇	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂⁵×C₆₇ ◁ G

4032 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 4032 conjugacy classes of subgroups — too many to draw. Only its 370 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 — 4 generators on 87 points:

(1,2)(3,4)(5,6,9,12)(7,10,14,17)(8,11,15,18)(13,19,20,16)(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);(5,7)(6,10)(8,13)(9,14)(11,16)(12,17)(15,20)(18,19);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,20)(17,19);(2,4)

Conjugacy classes & character table

3283 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 25888);