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G = D15⋊C8order 240 = 24·3·5

The semidirect product of D15 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D15⋊C8, D30.1C4, Dic3.2F5, Dic5.8D6, C5⋊C83S3, C51(S3×C8), C152(C2×C8), C31(D5⋊C8), C15⋊C82C2, C2.3(S3×F5), C6.5(C2×F5), C10.5(C4×S3), C30.5(C2×C4), D30.C2.2C2, (C5×Dic3).1C4, (C3×Dic5).8C22, (C3×C5⋊C8)⋊2C2, SmallGroup(240,99)

Series: Derived Chief Lower central Upper central

C1C15 — D15⋊C8
C1C5C15C30C3×Dic5C3×C5⋊C8 — D15⋊C8
C15 — D15⋊C8
C1C2

Generators and relations for D15⋊C8
 G = < a,b,c | a15=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a12b >

15C2
15C2
3C4
5C4
15C22
5S3
5S3
3D5
3D5
5C8
15C2×C4
15C8
5C12
5D6
3D10
3C20
15C2×C8
5C4×S3
5C24
5C3⋊C8
3C5⋊C8
3C4×D5
5S3×C8
3D5⋊C8

Character table of D15⋊C8

 class 12A2B2C34A4B4C4D568A8B8C8D8E8F8G8H1012A12B1520A20B24A24B24C24D30
 size 11151523355425555151515154101081212101010108
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1111111-1-1-1-11    linear of order 2
ρ311-1-11-1-111111111-1-1-1-11111-1-111111    linear of order 2
ρ411-1-11-1-11111-1-1-1-111111111-1-1-1-1-1-11    linear of order 2
ρ511-1-1111-1-111-ii-ii-ii-ii1-1-1111-i-iii1    linear of order 4
ρ611-1-1111-1-111i-ii-ii-ii-i1-1-1111ii-i-i1    linear of order 4
ρ711111-1-1-1-111-ii-iii-ii-i1-1-11-1-1-i-iii1    linear of order 4
ρ811111-1-1-1-111i-ii-i-ii-ii1-1-11-1-1ii-i-i1    linear of order 4
ρ91-11-11i-ii-i1-1ζ83ζ8ζ87ζ85ζ85ζ83ζ8ζ87-1-ii1i-iζ83ζ87ζ8ζ85-1    linear of order 8
ρ101-11-11-ii-ii1-1ζ8ζ83ζ85ζ87ζ87ζ8ζ83ζ85-1i-i1-iiζ8ζ85ζ83ζ87-1    linear of order 8
ρ111-1-111-iii-i1-1ζ87ζ85ζ83ζ8ζ85ζ83ζ8ζ87-1-ii1-iiζ87ζ83ζ85ζ8-1    linear of order 8
ρ121-11-11-ii-ii1-1ζ85ζ87ζ8ζ83ζ83ζ85ζ87ζ8-1i-i1-iiζ85ζ8ζ87ζ83-1    linear of order 8
ρ131-11-11i-ii-i1-1ζ87ζ85ζ83ζ8ζ8ζ87ζ85ζ83-1-ii1i-iζ87ζ83ζ85ζ8-1    linear of order 8
ρ141-1-111i-i-ii1-1ζ8ζ83ζ85ζ87ζ83ζ85ζ87ζ8-1i-i1i-iζ8ζ85ζ83ζ87-1    linear of order 8
ρ151-1-111-iii-i1-1ζ83ζ8ζ87ζ85ζ8ζ87ζ85ζ83-1-ii1-iiζ83ζ87ζ8ζ85-1    linear of order 8
ρ161-1-111i-i-ii1-1ζ85ζ87ζ8ζ83ζ87ζ8ζ83ζ85-1i-i1i-iζ85ζ8ζ87ζ83-1    linear of order 8
ρ172200-100222-1-2-2-2-200002-1-1-1001111-1    orthogonal lifted from D6
ρ182200-100222-1222200002-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ192200-100-2-22-1-2i2i-2i2i0000211-100ii-i-i-1    complex lifted from C4×S3
ρ202200-100-2-22-12i-2i2i-2i0000211-100-i-iii-1    complex lifted from C4×S3
ρ212-200-1002i-2i2183887850000-2i-i-100ζ87ζ83ζ85ζ81    complex lifted from S3×C8
ρ222-200-100-2i2i2188385870000-2-ii-100ζ85ζ8ζ87ζ831    complex lifted from S3×C8
ρ232-200-1002i-2i2187858380000-2i-i-100ζ83ζ87ζ8ζ851    complex lifted from S3×C8
ρ242-200-100-2i2i2185878830000-2-ii-100ζ8ζ85ζ83ζ871    complex lifted from S3×C8
ρ25440044400-1400000000-100-1-1-10000-1    orthogonal lifted from F5
ρ2644004-4-400-1400000000-100-1110000-1    orthogonal lifted from C2×F5
ρ274-40044i-4i00-1-400000000100-1-ii00001    complex lifted from D5⋊C8, Schur index 2
ρ284-4004-4i4i00-1-400000000100-1i-i00001    complex lifted from D5⋊C8, Schur index 2
ρ298-800-40000-24000000002001000000-1    orthogonal faithful, Schur index 2
ρ308800-40000-2-400000000-20010000001    orthogonal lifted from S3×F5

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

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

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

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

D15⋊C8 is a maximal subgroup of   S3×D5⋊C8  D60.C4  D15⋊M4(2)  Dic6.F5  C5⋊C8.D6  D15⋊C8⋊C2  D152M4(2)
D15⋊C8 is a maximal quotient of   D15⋊C16  D30.C8  Dic3×C5⋊C8  D30⋊C8  Dic15⋊C8

Matrix representation of D15⋊C8 in GL6(𝔽241)

24010000
24000000
000010
000001
00240240240240
001000
,
24000000
24010000
00002400
00024000
00240000
001111
,
24000000
02400000
001410138138
001381380141
0010331030
00100238238100

G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240,0],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,141,138,103,100,0,0,0,138,3,238,0,0,138,0,103,238,0,0,138,141,0,100] >;

D15⋊C8 in GAP, Magma, Sage, TeX

D_{15}\rtimes C_8
% in TeX

G:=Group("D15:C8");
// GroupNames label

G:=SmallGroup(240,99);
// by ID

G=gap.SmallGroup(240,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,50,490,3461,1745]);
// Polycyclic

G:=Group<a,b,c|a^15=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations

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Subgroup lattice of D15⋊C8 in TeX
Character table of D15⋊C8 in TeX

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