C₆₇×#32869 #30712120

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#32869

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

562 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 17152

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

(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,24,28,29)(21,30,22,25)(26,31,32,27)(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)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,23,28)(18,24,20,29)(21,27,22,31)(25,32,30,26);(1,3)(2,6)(4,8)(5,7)(10,13)(11,15)(17,20,23,18)(19,26,28,32)(21,30,22,25)(24,31,29,27);(3,8)(5,6)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,20,30)(19,27,28,31)(24,32,29,26)

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 25857);