C₆₇×#11521 #30690772

order	17152
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₇×#11521

Subgroups

	subgroup	G/H
center	C₄×C₆₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₆₇
socle	C₂×C₆₇	#1464
prime radical	C₂⁴×C₆₇	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

330 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

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, 4509);