C₆₇×#13719 #30692970

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₇×#13719

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
prime radical	C₆₇×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

202 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 8576

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

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

Conjugacy classes & character table

4690 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6707);