#30627803

order	15655
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃₁×C₁₀₁ ⋊ C₅
Sylow	C₁₀₁ ⋊ ΓL(1,𝔽₃₂)
Sylow	C₃₁ ⋊ C₁₀₁⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₁×C₁₀₁	C₅
LCS	G ▷ C₃₁×C₁₀₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 3131

A minimal faithful permutation representation — 2 generators on 132 points:

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

Polycyclic presentation:

5,31,101|b^a=b^16,c^a=c^36

Conjugacy classes & character table

631 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#30627803, #44531126, #44531132, …
maximal normal subgroup of	C₂×#30627803, #44531126, #44531132, …
maximal quotient of	C₂×#30627803

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15655, 6);