C₂×#568×C₁₃₇ #31778583

order	17536
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#568×C₁₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂⁴
commutator	C₂	C₂⁴×C₄×C₁₃₇
Frattini, cosocle	C₄	C₂⁵×C₁₃₇
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	C₂²×#41×C₁₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ G

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 155
transitive degree	μ_t(G) = 4384

A minimal faithful permutation representation — 5 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);(3,4,6,10,7,9,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,14)(10,15)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18);(4,9)(8,14)(10,16)(15,18)

Conjugacy classes & character table

9316 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃₇×#10742, C₁₃₇×#10744, C₁₃₇×#10746, …
maximal normal subgroup of	C₁₃₇×#10742, C₁₃₇×#10744, C₁₃₇×#10746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17536, 2304);

C2×#568×C137 #31778583