C₆₇×#7257 #30686508

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#7257

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#327
commutator	C₂×C₈	C₄²×C₆₇
Frattini, cosocle	C₂²×M₁₆	C₂²×C₆₇
socle	C₂×C₆₇	#1294
prime radical	C₂⁴×C₆₇	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

302 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 2 generators on 99 points:

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

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, 245);