C₆₇×#7357 #30686608

order	17152
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₇×#7357

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#353
commutator	C₄²	C₂×C₈×C₆₇
Frattini, cosocle	#423	C₂²×C₆₇
socle	C₂×C₆₇	#1318
prime radical	#532×C₆₇	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #423×C₆₇ ◁ G

222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Conjugacy classes & character table

2278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 345);