#1271×C₁₃₇ #31776282

order	17536
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1271×C₁₃₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂²×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

158 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 161
transitive degree	μ_t(G) = 17536

A minimal faithful permutation representation — 2 generators on 161 points:

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

Conjugacy classes & character table

5206 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#1271×C₁₃₇, C₁₃₇×#7560, C₁₃₇×#7562, …
maximal normal subgroup of	C₂×#1271×C₁₃₇, C₁₃₇×#7560, C₁₃₇×#7562, …
maximal quotient of	C₁₃₇×#7025, C₁₃₇×#7118, C₁₃₇×#7119, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 3);