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G = C8⋊D15  order 240 = 24·3·5

2nd semidirect product of C8 and D15 acting via D15/C15=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C40⋊2S3, C8⋊2D15, C24⋊2D5, C120⋊2C2, C4.8D30, C2.3D60, C6.1D20, C15⋊7SD16, D60.1C2, C30.19D4, C20.43D6, C10.1D12, Dic30⋊1C2, C12.43D10, C60.50C22, C5⋊1(C24⋊C2), C3⋊1(C40⋊C2), SmallGroup(240,67)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C60 — C8⋊D15
Chief series
C1 — C5 — C15 — C30 — C60 — D60 — C8⋊D15
Lower central
C15 — C30 — C60 — C8⋊D15
Upper central
C1 — C2 — C4 — C8

Generators and relations for C8⋊D15
 G = < a,b,c | a8=b15=c2=1, ab=ba, cac=a3, cbc=b-1 >

C1
C2
60C2
C3
C5
C4
30C22
30C4
C6
20S3
C10
12D5
C15
C8
15Q8
15D4
C12
10Dic3
10D6
C20
6D10
6Dic5
C30
4D15
15SD16
C24
5Dic6
5D12
C40
3D20
3Dic10
C60
2D30
2Dic15
5C24⋊C2
3C40⋊C2
D60
Dic30
C120
C8⋊D15

Smallest permutation representation of C8⋊D15
►On 120 points
Generators in S120
(1 118 47 88 28 103 35 62)(2 119 48 89 29 104 36 63)(3 120 49 90 30 105 37 64)(4 106 50 76 16 91 38 65)(5 107 51 77 17 92 39 66)(6 108 52 78 18 93 40 67)(7 109 53 79 19 94 41 68)(8 110 54 80 20 95 42 69)(9 111 55 81 21 96 43 70)(10 112 56 82 22 97 44 71)(11 113 57 83 23 98 45 72)(12 114 58 84 24 99 31 73)(13 115 59 85 25 100 32 74)(14 116 60 86 26 101 33 75)(15 117 46 87 27 102 34 61)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 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)

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

63 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12234455688101012121515151520202020242424243030303040···4060···60120···120
size1160226022222222222222222222222222···22···22···2

63 irreducible representations

dim111122222222222222
type++++++++++++++
imageC1C2C2C2S3D4D5D6SD16D10D12D15D20C24⋊C2D30C40⋊C2D60C8⋊D15
kernelC8⋊D15C120Dic30D60C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps1111112122244448816

Matrix representation of C8⋊D15 ►in GL2(𝔽241) generated by

13257
184147
,
94110
131161
,
1190
0240
G:=sub<GL(2,GF(241))| [132,184,57,147],[94,131,110,161],[1,0,190,240] >;
 

C8⋊D15 in GAP, Magma, Sage, TeX 

C_8\rtimes D_{15}
 
% in TeX
 
G:=Group("C8:D15");
 
// GroupNames label
 
G:=SmallGroup(240,67);
 
// by ID
 
G=gap.SmallGroup(240,67);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,31,218,50,964,6917]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^15=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
 
// generators/relations
 

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