C₆₇×#32874 #30712125

order	17152
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₆₇×#32874

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂⁴×C₆₇ ◁ G

688 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 99
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 4 generators on 99 points:

(1,2)(3,11)(4,15)(5,9)(6,10)(7,14)(8,17)(12,21)(13,24)(16,29)(18,23)(19,26)(20,28)(22,30)(25,32)(27,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,21,15,19)(9,20,23,24)(16,27)(17,26)(18,28)(22,31,29,32)(25,30);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,21)(9,22,23,29)(13,27)(14,26)(18,30)(20,32,24,31)(25,28);(1,5,6,18)(2,9,10,23)(3,13,14,28)(4,16,17,30)(7,20,11,24)(8,22,15,29)(12,25,26,27)(19,31,21,32)

Conjugacy classes & character table

1876 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 25862);