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## G = C15⋊D8order 240 = 24·3·5

### 1st semidirect product of C15 and D8 acting via D8/C4=C22

Aliases: C151D8, D122D5, D202S3, C20.1D6, C30.1D4, C12.1D10, C60.22C22, C52(D4⋊S3), C32(D4⋊D5), C153C86C2, (C5×D12)⋊4C2, (C3×D20)⋊4C2, C4.15(S3×D5), C6.7(C5⋊D4), C10.7(C3⋊D4), C2.4(C15⋊D4), SmallGroup(240,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C15⋊D8
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C15⋊D8
 Lower central C15 — C30 — C60 — C15⋊D8
 Upper central C1 — C2 — C4

Generators and relations for C15⋊D8
G = < a,b,c | a15=b8=c2=1, bab-1=a-1, cac=a4, cbc=b-1 >

Character table of C15⋊D8

 class 1 2A 2B 2C 3 4 5A 5B 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 12 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D size 1 1 12 20 2 2 2 2 2 20 20 30 30 2 2 12 12 12 12 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 2 -1 2 2 2 -1 -1 -1 0 0 2 2 0 0 0 0 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 -2 -1 2 2 2 -1 1 1 0 0 2 2 0 0 0 0 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 2 0 0 2 -2 2 2 2 0 0 0 0 2 2 0 0 0 0 -2 2 2 -2 -2 2 2 -2 -2 -2 -2 orthogonal lifted from D4 ρ8 2 2 2 0 2 2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 2 -2 0 2 2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ10 2 2 2 0 2 2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 -2 0 0 2 0 2 2 -2 0 0 √2 -√2 -2 -2 0 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 -2 0 2 2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ13 2 -2 0 0 2 0 2 2 -2 0 0 -√2 √2 -2 -2 0 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 0 0 -1 -2 2 2 -1 -√-3 √-3 0 0 2 2 0 0 0 0 1 -1 -1 -2 -2 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 0 2 -2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 -2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 2 0 0 -1 -2 2 2 -1 √-3 -√-3 0 0 2 2 0 0 0 0 1 -1 -1 -2 -2 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ17 2 2 0 0 2 -2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ18 2 2 0 0 2 -2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 complex lifted from C5⋊D4 ρ19 2 2 0 0 2 -2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 -2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ20 4 -4 0 0 -2 0 4 4 2 0 0 0 0 -4 -4 0 0 0 0 0 -2 -2 0 0 2 2 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ21 4 4 0 0 -2 4 -1-√5 -1+√5 -2 0 0 0 0 -1+√5 -1-√5 0 0 0 0 -2 1+√5/2 1-√5/2 -1-√5 -1+√5 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ22 4 -4 0 0 4 0 -1-√5 -1+√5 -4 0 0 0 0 1-√5 1+√5 0 0 0 0 0 -1-√5 -1+√5 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ23 4 -4 0 0 4 0 -1+√5 -1-√5 -4 0 0 0 0 1+√5 1-√5 0 0 0 0 0 -1+√5 -1-√5 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ24 4 4 0 0 -2 4 -1+√5 -1-√5 -2 0 0 0 0 -1-√5 -1+√5 0 0 0 0 -2 1-√5/2 1+√5/2 -1+√5 -1-√5 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ25 4 4 0 0 -2 -4 -1-√5 -1+√5 -2 0 0 0 0 -1+√5 -1-√5 0 0 0 0 2 1+√5/2 1-√5/2 1+√5 1-√5 1+√5/2 1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ26 4 4 0 0 -2 -4 -1+√5 -1-√5 -2 0 0 0 0 -1-√5 -1+√5 0 0 0 0 2 1-√5/2 1+√5/2 1-√5 1+√5 1-√5/2 1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ27 4 -4 0 0 -2 0 -1-√5 -1+√5 2 0 0 0 0 1-√5 1+√5 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 complex faithful ρ28 4 -4 0 0 -2 0 -1-√5 -1+√5 2 0 0 0 0 1-√5 1+√5 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 complex faithful ρ29 4 -4 0 0 -2 0 -1+√5 -1-√5 2 0 0 0 0 1+√5 1-√5 0 0 0 0 0 1-√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 complex faithful ρ30 4 -4 0 0 -2 0 -1+√5 -1-√5 2 0 0 0 0 1+√5 1-√5 0 0 0 0 0 1-√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 complex faithful

Smallest permutation representation of C15⋊D8
On 120 points
Generators in S120
(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 62 35 120 17 78 50 93)(2 61 36 119 18 77 51 92)(3 75 37 118 19 76 52 91)(4 74 38 117 20 90 53 105)(5 73 39 116 21 89 54 104)(6 72 40 115 22 88 55 103)(7 71 41 114 23 87 56 102)(8 70 42 113 24 86 57 101)(9 69 43 112 25 85 58 100)(10 68 44 111 26 84 59 99)(11 67 45 110 27 83 60 98)(12 66 31 109 28 82 46 97)(13 65 32 108 29 81 47 96)(14 64 33 107 30 80 48 95)(15 63 34 106 16 79 49 94)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(16 28)(18 21)(19 25)(20 29)(23 26)(24 30)(31 49)(32 53)(33 57)(34 46)(35 50)(36 54)(37 58)(38 47)(39 51)(40 55)(41 59)(42 48)(43 52)(44 56)(45 60)(61 104)(62 93)(63 97)(64 101)(65 105)(66 94)(67 98)(68 102)(69 91)(70 95)(71 99)(72 103)(73 92)(74 96)(75 100)(76 112)(77 116)(78 120)(79 109)(80 113)(81 117)(82 106)(83 110)(84 114)(85 118)(86 107)(87 111)(88 115)(89 119)(90 108)

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

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

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

C15⋊D8 is a maximal subgroup of
C405D6  D246D5  C408D6  Dic6.D10  D20.34D6  C60.36D4  D2021D6  D5×D4⋊S3  S3×D4⋊D5  D1210D10  D2010D6  D20⋊D6  D12⋊D10  D20.14D6  D20.27D6
C15⋊D8 is a maximal quotient of
C15⋊D16  C40.D6  C15⋊SD32  C15⋊Q32  C30.D8  D12⋊Dic5  C30.20D8

Matrix representation of C15⋊D8 in GL6(𝔽241)

 225 0 0 0 0 0 225 15 0 0 0 0 0 0 190 51 0 0 0 0 190 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 17 0 0 0 0 170 1 0 0 0 0 0 0 190 51 0 0 0 0 1 51 0 0 0 0 0 0 0 22 0 0 0 0 230 22
,
 1 0 0 0 0 0 71 240 0 0 0 0 0 0 51 190 0 0 0 0 240 190 0 0 0 0 0 0 1 0 0 0 0 0 1 240

G:=sub<GL(6,GF(241))| [225,225,0,0,0,0,0,15,0,0,0,0,0,0,190,190,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,170,0,0,0,0,17,1,0,0,0,0,0,0,190,1,0,0,0,0,51,51,0,0,0,0,0,0,0,230,0,0,0,0,22,22],[1,71,0,0,0,0,0,240,0,0,0,0,0,0,51,240,0,0,0,0,190,190,0,0,0,0,0,0,1,1,0,0,0,0,0,240] >;

C15⋊D8 in GAP, Magma, Sage, TeX

C_{15}\rtimes D_8
% in TeX

G:=Group("C15:D8");
// GroupNames label

G:=SmallGroup(240,13);
// by ID

G=gap.SmallGroup(240,13);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,218,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^4,c*b*c=b^-1>;
// generators/relations

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