#49138773

order	36414
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁₇

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₁₇² ⋊ C₂×C₉
Hall	C₇⋊#14579585 ⋊ C₂
Hall	C₇⋊(C₁₇⋊D₁₇) ⋊ C₉
Sylow	C₁₇² ⋊ D₇⋊C₉
Sylow	C₇ ⋊ #19104523

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₇²	C₂×C₉
prime radical	#27131472	C₃
cosocle	C₇×C₁₇²⋊C₃	C₂×C₃
LCS	G ▷ C₇×C₁₇²
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 2 generators on 160 points:

(2,5,6,3,4,7)(8,13,54,10,26,12,16,9,23)(11,37,155,17,110,104,49,45,107,19,47,146,21,91,46,38,20,90)(14,57,51,72,139,52,76,24,93,29,74,22,31,122,39,131,30,113)(15,123,115,78,140,96,81,58,97,32,80,144,34,117,40,136,33,111)(18,87,62,36,48,119,101,82,151,41,100,109,43,86,99,88,42,143)(25,112,79,55,147,77,125,50,63,59,92,73,61,152,83,118,60,64)(27,116,141,127,85,150,35,126,159,65,129,98,67,75,128,84,66,156)(28,120,102,133,124,103,135,56,145,68,134,44,70,142,89,138,69,153)(53,71,157,94,158,154,149,105,121,114,137,160,95,132,108,148,106,130);(1,2,4,7,6,5,3)(8,11,22,53,115,101,152,145,159,156,153,147,88,144,114,51,19)(9,14,32,79,84,135,48,106,20,45,105,86,138,35,73,15,29)(10,17,39,94,96,41,60,44,98,141,102,50,18,40,95,52,21)(12,24,33,59,85,142,109,154,104,46,108,62,124,75,25,58,30)(13,27,37,28,31,77,71,42,78,34,82,137,83,72,68,47,65)(23,55,119,107,69,123,57,66,130,121,126,74,80,56,90,99,61)(26,63,100,91,133,81,76,127,158,132,67,131,136,70,110,87,64)(54,116,155,120,122,125,157,143,140,117,151,160,118,139,134,146,129)

Polycyclic presentation:

2,3^2,7,17^2|b^3=c,d^a=d^6,e^a=e^16,f^a=f^16,d^b=d^2,e^b=e^9f^9,f^b=e^13f^15,e^c=f^7,f^c=e^12f^16

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 68);