C₆₇×#32868 #30712119

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

Aliases

direct product

C₆₇×#32868

Subgroups

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

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

normal subgroups only

This group has 596 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) = 8576

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

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, 25856);