C₁₅₁×#12728 #50317653

order	38656
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₅₁×#12728

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₅₁	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂²×D₄	C₂³×C₁₅₁
prime radical	#535×C₁₅₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₅₁ ◁ C₂³×C₁₅₁ ◁ #518×C₁₅₁ ◁ G

714 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 183
transitive degree	μ_t(G) = 4832

A minimal faithful permutation representation — 3 generators on 183 points:

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

Conjugacy classes & character table

5134 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 5716);