#2002×C₁₃₇ #31777013

order	17536
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#2002×C₁₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

478 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 149
transitive degree	μ_t(G) = 2192

A minimal faithful permutation representation — 3 generators on 149 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#2002×C₁₃₇, C₁₃₇×#23399, C₁₃₇×#23400, …
maximal normal subgroup of	C₂×#2002×C₁₃₇, C₁₃₇×#23399, C₁₃₇×#23400, …
maximal quotient of	C₁₃₇×#9914, C₁₃₇×#9920, C₁₃₇×#9922, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 734);