#1819×C₁₃₇ #31776830

order	17536
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#1819×C₁₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂⁴	C₂³×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂⁵×C₁₃₇ ◁ G

504 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 153
transitive degree	μ_t(G) = 2192

A minimal faithful permutation representation — 3 generators on 153 points:

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

Conjugacy classes & character table

4384 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃₇×#7213, C₁₃₇×#7349, C₁₃₇×#8535, …
maximal normal subgroup of	C₁₃₇×#7213, C₁₃₇×#7349, C₁₃₇×#8535, …
maximal quotient of	C₁₃₇×#8698, C₁₃₇×#8700, C₁₃₇×#8702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 551);