C₂×#360×C₁₃₇ #31777155

order	17536
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#360×C₁₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#102
commutator	C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₃₇
prime radical	C₂×#135×C₁₃₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ G

232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 155
transitive degree	μ_t(G) = 4384

A minimal faithful permutation representation — 3 generators on 155 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×#360×C₁₃₇, C₁₃₇×#12650, C₁₃₇×#12735, …
maximal normal subgroup of	C₄×#360×C₁₃₇, C₁₃₇×#12650, C₁₃₇×#12735, …
maximal quotient of	C₂×#1333×C₁₃₇, C₂×#1334×C₁₃₇, C₁₃₇×#11598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 876);