C₆₇×#11395 #30690646

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₇×#11395

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#393
commutator	C₂⁴	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂×C₆₇	#1446
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

820 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 3 generators on 99 points:

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

Conjugacy classes & character table

3082 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 4383);