C₆₇×#10668 #30689919

order	17152
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#10668

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#399
prime radical	C₂³×C₆₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

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

normal subgroups only

This group has 316 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:

(1,2,3,7,5,8,4,6)(9,10,13,18,14,19,15,29)(11,20,22,32,24,36,23,16)(12,25,27,35,28,17,30,34)(21,37,26,40,38,33,39,31)(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);(2,6)(3,4)(7,8)(9,11,15,22,14,24,13,23)(10,16,29,20,19,32,18,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(1,2,5,8)(3,6,4,7)(9,12,14,28)(10,17,19,25)(11,21,24,38)(13,27,15,30)(16,31,32,40)(18,34,29,35)(20,37,36,33)(22,26,23,39)

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, 3656);