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G = C15⋊D8order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C60 — C15⋊D8
C1C5C15C30C60C3×D20 — C15⋊D8
C15C30C60 — C15⋊D8
C1C2C4

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

12C2
20C2
6C22
10C22
4S3
20C6
4D5
12C10
3D4
5D4
15C8
2D6
10C2×C6
2D10
6C2×C10
4C5×S3
4C3×D5
15D8
5C3×D4
5C3⋊C8
3C5×D4
3C52C8
2C6×D5
2S3×C10
5D4⋊S3
3D4⋊D5

Character table of C15⋊D8

 class 12A2B2C345A5B6A6B6C8A8B10A10B10C10D10E10F1215A15B20A20B30A30B60A60B60C60D
 size 1112202222220203030221212121244444444444
ρ1111111111111111111111111111111    trivial
ρ211-1-111111-1-11111-1-1-1-111111111111    linear of order 2
ρ3111-111111-1-1-1-111111111111111111    linear of order 2
ρ411-111111111-1-111-1-1-1-111111111111    linear of order 2
ρ52202-1222-1-1-100220000-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ6220-2-1222-11100220000-1-1-122-1-1-1-1-1-1    orthogonal lifted from D6
ρ722002-22220000220000-222-2-222-2-2-2-2    orthogonal lifted from D4
ρ8222022-1+5/2-1-5/220000-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/22-1+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
ρ922-2022-1+5/2-1-5/220000-1-5/2-1+5/21-5/21+5/21+5/21-5/22-1+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
ρ10222022-1-5/2-1+5/220000-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/22-1-5/2-1+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
ρ112-2002022-2002-2-2-2000002200-2-20000    orthogonal lifted from D8
ρ1222-2022-1-5/2-1+5/220000-1+5/2-1-5/21+5/21-5/21-5/21+5/22-1-5/2-1+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
ρ132-2002022-200-22-2-2000002200-2-20000    orthogonal lifted from D8
ρ142200-1-222-1--3-3002200001-1-1-2-2-1-11111    complex lifted from C3⋊D4
ρ1522002-2-1-5/2-1+5/220000-1+5/2-1-5/2ζ5352ζ5455455352-2-1-5/2-1+5/21+5/21-5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2    complex lifted from C5⋊D4
ρ162200-1-222-1-3--3002200001-1-1-2-2-1-11111    complex lifted from C3⋊D4
ρ1722002-2-1+5/2-1-5/220000-1-5/2-1+5/2545ζ53525352ζ545-2-1+5/2-1-5/21-5/21+5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2    complex lifted from C5⋊D4
ρ1822002-2-1-5/2-1+5/220000-1+5/2-1-5/25352545ζ545ζ5352-2-1-5/2-1+5/21+5/21-5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2    complex lifted from C5⋊D4
ρ1922002-2-1+5/2-1-5/220000-1-5/2-1+5/2ζ5455352ζ5352545-2-1+5/2-1-5/21-5/21+5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2    complex lifted from C5⋊D4
ρ204-400-204420000-4-400000-2-200220000    orthogonal lifted from D4⋊S3, Schur index 2
ρ214400-24-1-5-1+5-20000-1+5-1-50000-21+5/21-5/2-1-5-1+51+5/21-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from S3×D5
ρ224-40040-1-5-1+5-400001-51+500000-1-5-1+5001+51-50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ234-40040-1+5-1-5-400001+51-500000-1+5-1-5001-51+50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ244400-24-1+5-1-5-20000-1-5-1+50000-21-5/21+5/2-1+5-1-51-5/21+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from S3×D5
ρ254400-2-4-1-5-1+5-20000-1+5-1-5000021+5/21-5/21+51-51+5/21-5/2-1-5/2-1-5/2-1+5/2-1+5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ264400-2-4-1+5-1-5-20000-1-5-1+5000021-5/21+5/21-51+51-5/21+5/2-1+5/2-1+5/2-1-5/2-1-5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ274-400-20-1-5-1+5200001-51+5000001+5/21-5/200-1-5/2-1+5/2-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ5243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ54ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5    complex faithful
ρ284-400-20-1-5-1+5200001-51+5000001+5/21-5/200-1-5/2-1+5/2-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ524ζ3ζ54-2ζ4ζ3ζ54ζ544ζ543ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5    complex faithful
ρ294-400-20-1+5-1-5200001+51-5000001-5/21+5/200-1+5/2-1-5/24ζ3ζ54-2ζ4ζ3ζ54ζ544ζ543ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52    complex faithful
ρ304-400-20-1+5-1-5200001+51-5000001-5/21+5/200-1+5/2-1-5/243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ54ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52    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 64 35 110 17 90 54 99)(2 63 36 109 18 89 55 98)(3 62 37 108 19 88 56 97)(4 61 38 107 20 87 57 96)(5 75 39 106 21 86 58 95)(6 74 40 120 22 85 59 94)(7 73 41 119 23 84 60 93)(8 72 42 118 24 83 46 92)(9 71 43 117 25 82 47 91)(10 70 44 116 26 81 48 105)(11 69 45 115 27 80 49 104)(12 68 31 114 28 79 50 103)(13 67 32 113 29 78 51 102)(14 66 33 112 30 77 52 101)(15 65 34 111 16 76 53 100)
(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 53)(32 57)(33 46)(34 50)(35 54)(36 58)(37 47)(38 51)(39 55)(40 59)(41 48)(42 52)(43 56)(44 60)(45 49)(61 102)(62 91)(63 95)(64 99)(65 103)(66 92)(67 96)(68 100)(69 104)(70 93)(71 97)(72 101)(73 105)(74 94)(75 98)(76 114)(77 118)(78 107)(79 111)(80 115)(81 119)(82 108)(83 112)(84 116)(85 120)(86 109)(87 113)(88 117)(89 106)(90 110)

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

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

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

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)

22500000
225150000
001905100
0019024000
000010
000001
,
240170000
17010000
001905100
0015100
0000022
000023022
,
100000
712400000
005119000
0024019000
000010
00001240

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

Export

Subgroup lattice of C15⋊D8 in TeX
Character table of C15⋊D8 in TeX

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