C₆₇×#7390 #30686641

order	17152
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7390

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#1333
commutator	C₁₆	C₂×C₈×C₆₇
Frattini, cosocle	#346	C₂²×C₆₇
prime radical	C₂×D₁₆×C₆₇	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄×C₈×C₆₇ ◁ #346×C₆₇ ◁ G

128 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

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, 378);