C₁₁×C₂₄₁⋊C₅ #27210420

order	13255
automorphism	?
homology	Abelianization (H₁) = C₅×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×C₂₄₁⋊C₅
unambiguous semidirect product	C₂₄₁⋊(C₅×C₁₁)

Splittings

Fitting, socle, Hall	C₁₁×C₂₄₁ ⋊ C₅
derived, Sylow	C₂₄₁ ⋊ C₅×C₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	C₂₄₁⋊C₅
commutator, cosocle, hypocenter	C₂₄₁	C₅×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁×C₂₄₁	C₅
LCS	G ▷ C₂₄₁
UCS	C₁ ◁ C₁₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 252
transitive degree	μ_t(G) = 2651

A minimal faithful permutation representation — 2 generators on 252 points:

(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,55,75)(10,19,36,18,34)(12,23,44,80,88)(14,27,52,94,63)(16,31,59,105,165)(20,38,70,49,89)(22,42,76,73,64)(24,46,84,140,127)(26,50,91,35,65)(28,54,97,60,107)(30,57,102,155,113)(32,61,109,170,176)(37,68,41,74,67)(39,72,124,99,130)(43,78,131,134,147)(45,82,137,153,114)(47,86,144,171,216)(51,92,151,90,148)(53,96,62,111,115)(56,100,136,66,116)(58,104,163,213,168)(69,120,81,135,150)(71,122,95,154,118)(77,129,128,126,119)(79,133,189,196,186)(83,139,194,108,169)(85,142,199,209,177)(87,146,205,234,221)(93,152,173,158,188)(98,157,145,203,178)(101,159,198,149,206)(103,162,210,174,179)(106,166,125,117,161)(110,172,123,184,156)(112,175,181,138,193)(121,182,141,197,191)(132,187,192,208,180)(143,201,229,236,218)(160,212,231,211,225)(164,215,237,232,222)(167,217,190,207,227)(183,214,202,219,185)(195,226,204,233,223)(200,228,235,220,224)(230,239,241,240,238)(242,243,244,245,246,247,248,249,250,251,252);(1,2,4,8,16,32,62,112,176,221,154,208,233,240,239,52,95,155,209,50,29,56,101,160,27,53,80,134,23,45,83,70,15,30,58,68,119,180,217,65,115,178,91,150,172,219,89,147,169,218,226,21,41,75,127,19,37,69,121,183,146,170,40,73,111,174,163,214,237,241,162,144,202,231,159,142,200,199,227,139,195,197,189,133,190,225,187,5,10,20,39,3,6,12,24,47,87,105,17,33,63,113,97,156,210,235,96,59,106,167,92,57,103,137,192,54,98,135,124,46,85,143,122,31,60,108,120,181,223,151,116,179,222,55,99,158,211,148,44,81,136,191,42,77,130,186,38,71,123,76,128,165,216,74,126,175,220,215,205,234,94,153,203,232,229,212,201,230,228,25,49,90,149,207,13,26,51,93,11,22,43,79,7,14,28,36,67,118,166,34,64,114,177,157,109,171,48,88,107,168,194,152,104,164,193,9,18,35,66,117,86,145,204,184,61,110,173,182,224,238,102,161,213,236,100,84,141,198,206,82,138,140,196,78,132,188,131,72,125,185,129)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₁×C₂₄₁⋊C₅, C₁₁×D₂₄₁⋊C₅, D₁₁×C₂₄₁⋊C₅, …
maximal normal subgroup of	C₂×C₁₁×C₂₄₁⋊C₅, C₁₁×D₂₄₁⋊C₅, D₁₁×C₂₄₁⋊C₅, …
maximal quotient of	C₂×C₁₁×C₂₄₁⋊C₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13255, 2);

C11×C241⋊C5 #27210420