C₆₇×#13681 #30692932

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#13681

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₆₇
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄⋊Q₈×C₆₇ ◁ G

398 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 83
transitive degree	μ_t(G) = 1072

A minimal faithful permutation representation — 3 generators on 83 points:

(2,7)(3,10)(4,13)(8,16)(9,11)(12,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(1,2)(3,11)(4,12)(5,8)(6,9)(7,10)(13,16)(14,15);(1,2)(3,12)(4,11)(5,15)(6,9)(7,13)(8,14)(10,16)

Conjugacy classes & character table

1675 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6669);