C₆₇×#10666 #30689917

order	17152
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#10666

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#399
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 107
transitive degree	μ_t(G) = 4288

A minimal faithful permutation representation — 3 generators on 107 points:

(1,2,6,7,5,3,4,8)(9,10,15,19,14,20,13,18)(11,21,29,16,24,32,23,31)(12,17,27,36,28,35,26,34)(22,37,25,40,39,33,38,30)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,3,5,2)(4,8,6,7)(9,11,13,29,14,24,15,23)(10,16,18,32,20,31,19,21)(12,25,26,39,28,38,27,22)(17,33,34,30,35,37,36,40);(2,7)(3,8)(4,6)(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,30)(18,34)(19,36)(20,35)(21,33)(23,38)(24,39)(25,29)(31,40)(32,37)

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, 3654);