C₁₃₇×#11463 #46860849

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃₇×#11463

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

306 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 185
transitive degree	μ_t(G) = 17536

A minimal faithful permutation representation — 3 generators on 185 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,27,22,28,36,45,23,25,37,41,20,26,34,43)(19,29,31,46,30,39,47,44,32,42,35,48,33,24,38,40)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,39)(26,42)(27,44)(28,29)(31,37)(34,47)(36,38)(40,41)(43,46)(45,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(24,39)(26,28)(27,41)(29,42)(40,44)(43,45)(46,48)

Conjugacy classes & character table

7124 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 4451);