#3241×C₁₃₇ #31778252

order	17536
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#3241×C₁₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
prime radical	#540×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ #117×C₁₃₇ ◁ G

332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 169
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 4 generators on 169 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(2,8)(6,9)(7,16)(14,15)(19,26)(22,30)(27,32)(29,31)

Conjugacy classes & character table

3562 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₁₃₇×#21843, C₁₃₇×#21872, C₁₃₇×#21992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 1973);

#3241×C137 #31778252