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## G = D8×D15order 480 = 25·3·5

### Direct product of D8 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D8×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — D4×D15 — D8×D15
 Lower central C15 — C30 — C60 — D8×D15
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8×D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1300 in 152 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, D15, D15, C30, C30, C2×D8, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, Dic15, C60, D30, D30, C2×C30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, S3×D8, C153C8, C120, C4×D15, D60, C157D4, D4×C15, C22×D15, D5×D8, C8×D15, D120, D4⋊D15, C15×D8, D4×D15, D8×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, D15, C2×D8, C22×D5, S3×D4, D30, D4×D5, S3×D8, C22×D15, D5×D8, D4×D15, D8×D15

Smallest permutation representation of D8×D15
On 120 points
Generators in S120
(1 95 39 61 20 106 54 84)(2 96 40 62 21 107 55 85)(3 97 41 63 22 108 56 86)(4 98 42 64 23 109 57 87)(5 99 43 65 24 110 58 88)(6 100 44 66 25 111 59 89)(7 101 45 67 26 112 60 90)(8 102 31 68 27 113 46 76)(9 103 32 69 28 114 47 77)(10 104 33 70 29 115 48 78)(11 105 34 71 30 116 49 79)(12 91 35 72 16 117 50 80)(13 92 36 73 17 118 51 81)(14 93 37 74 18 119 52 82)(15 94 38 75 19 120 53 83)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 76)(28 77)(29 78)(30 79)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)
(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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 12 15A 15B 15C 15D 20A 20B 24A 24B 30A 30B 30C 30D 30E ··· 30L 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 8 8 8 8 10 10 10 10 10 10 12 15 15 15 15 20 20 24 24 30 30 30 30 30 ··· 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 4 4 15 15 60 60 2 2 30 2 2 2 8 8 2 2 30 30 2 2 8 8 8 8 4 2 2 2 2 4 4 4 4 2 2 2 2 8 ··· 8 4 4 4 4 4 4 4 4 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 D15 D30 D30 S3×D4 D4×D5 S3×D8 D5×D8 D4×D15 D8×D15 kernel D8×D15 C8×D15 D120 D4⋊D15 C15×D8 D4×D15 C5×D8 Dic15 D30 C3×D8 C40 C5×D4 D15 C24 C3×D4 D8 C8 D4 C10 C6 C5 C3 C2 C1 # reps 1 1 1 2 1 2 1 1 1 2 1 2 4 2 4 4 4 8 1 2 2 4 4 8

Matrix representation of D8×D15 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 0 208 0 0 168 219
,
 1 0 0 0 0 1 0 0 0 0 22 208 0 0 168 219
,
 16 68 0 0 173 178 0 0 0 0 1 0 0 0 0 1
,
 30 64 0 0 178 211 0 0 0 0 240 0 0 0 0 240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,168,0,0,208,219],[1,0,0,0,0,1,0,0,0,0,22,168,0,0,208,219],[16,173,0,0,68,178,0,0,0,0,1,0,0,0,0,1],[30,178,0,0,64,211,0,0,0,0,240,0,0,0,0,240] >;

D8×D15 in GAP, Magma, Sage, TeX

D_8\times D_{15}
% in TeX

G:=Group("D8xD15");
// GroupNames label

G:=SmallGroup(480,875);
// by ID

G=gap.SmallGroup(480,875);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,346,185,80,2693,18822]);
// Polycyclic

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

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