C₆₇×#7520 #30686771

order	17152
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₇×#7520

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₆₇
Frattini, cosocle	#492	C₂²×C₆₇
prime radical	#127×C₆₇	D₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄²×C₆₇ ◁ #492×C₆₇ ◁ G

156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 83 points:

(1,2)(3,9,12,8)(4,13,14,7)(5,10)(6,11,16,15)(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);(1,3,5,12)(2,6,10,13)(4,14,15,11)(7,8,16,9)

Conjugacy classes & character table

1474 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 508);