C₆₇×#7125 #30686376

order	17152
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7125

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#327
commutator	C₂×C₈	C₂×C₈×C₆₇
Frattini, cosocle	C₂×#97	C₂²×C₆₇
socle	C₂×C₆₇	#1271
prime radical	C₂³×C₆₇	#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

182 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,11,13,24,14,28,19,31)(4,15,17,8)(5,10)(6,9)(7,12,21,29,22,30,23,16)(18,27,32,25)(20,26)(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)(2,7,9,22)(4,16)(5,19)(6,14)(8,24,15,31)(10,21,26,23)(11,27,28,25)(12,18)(13,20)(17,29)(30,32)

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, 113);