#49138779

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
Hall	#14579585 ⋊ D₇
Sylow	C₁₇² ⋊ C₉×D₇
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:

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

Polycyclic presentation:

2,3^2,7,17^2|b^3=c,d^a=d^6,e^a=e^16,f^a=f^16,e^b=e^14f^12,f^b=e^13f^10,e^c=e^2f^15,f^c=e^12f^14

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 74);