C₂×C₁₃⋊(D₁₂₁⋊C₅) #44538078

order	31460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₁₃⋊(D₁₂₁⋊C₅)
unambiguous semidirect product	(C₂×C₁₃)⋊(D₁₂₁⋊C₅)
unambiguous semidirect product	(C₂×C₁₂₁⋊C₅)⋊D₁₃
unambiguous semidirect product	(C₂×D₁₅₇₃)⋊C₅

Splittings

Fitting	C₂×C₁₃×C₁₂₁ ⋊ C₂×C₅
derived, Hall	C₁₃×C₁₂₁ ⋊ C₂²×C₅
Hall	C₁₃×C₁₂₁⋊C₅ ⋊ C₂²
Hall	C₂×D₁₅₇₃ ⋊ C₅
Hall	C₁₂₁⋊C₅ ⋊ C₂×D₁₃
Sylow	C₁₂₁ ⋊ C₂×C₅×D₁₃
Sylow	C₁₃ ⋊ C₂×D₁₂₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₁₃⋊(D₁₂₁⋊C₅)
commutator, cosocle, hypocenter	C₁₃×C₁₂₁	C₂²×C₅
Frattini	C₁₁	C₂×C₁₃⋊AGL(1,𝔽₁₁)
Fitting	C₂×C₁₃×C₁₂₁	C₂×C₅
socle	C₂×C₁₁×C₁₃	AGL(1,𝔽₁₁)
LCS	G ▷ C₁₃×C₁₂₁
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 3146

A minimal faithful permutation representation — 2 generators on 136 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31460, 18);

C2×C13⋊(D121⋊C5) #44538078