C₆₇×#13665 #30692916

order	17152
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₇×#13665

Subgroups

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

284 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 135
transitive degree	μ_t(G) = 8576

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2)(3,4)(5,6,11,16)(7,17,21,48)(8,22,27,46)(9,28,31,62)(10,32,30,15)(12,34,37,18)(13,38,41,25)(14,42,44,24)(19,50,51,67)(20,52,33,47)(23,43,57,68)(26,60,59,54)(29,39,63,66)(35,61,64,56)(36,49,45,55)(40,58,53,65)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(5,7,11,21)(6,12,16,37)(8,23,27,57)(9,29,31,63)(10,33,30,20)(13,39,41,66)(14,43,44,68)(15,45,32,36)(17,46,48,22)(18,38,34,25)(19,42,51,24)(26,61,59,56)(28,35,62,64)(40,67,53,50)(47,54,52,60)(49,58,55,65);(2,4)

Conjugacy classes & character table

3283 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 6653);