C₆₇×#19084 #30698335

order	17152
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₇×#19084

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

824 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 230 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:

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

Conjugacy classes & character table

2680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 12072);