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## G = D5×C3⋊C8order 240 = 24·3·5

### Direct product of D5 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D5×C3⋊C8
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×C3⋊C8
 Lower central C15 — D5×C3⋊C8
 Upper central C1 — C4

Generators and relations for D5×C3⋊C8
G = < a,b,c,d | a5=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D5×C3⋊C8
On 120 points
Generators in S120
(1 28 113 88 99)(2 29 114 81 100)(3 30 115 82 101)(4 31 116 83 102)(5 32 117 84 103)(6 25 118 85 104)(7 26 119 86 97)(8 27 120 87 98)(9 22 39 72 43)(10 23 40 65 44)(11 24 33 66 45)(12 17 34 67 46)(13 18 35 68 47)(14 19 36 69 48)(15 20 37 70 41)(16 21 38 71 42)(49 112 78 95 61)(50 105 79 96 62)(51 106 80 89 63)(52 107 73 90 64)(53 108 74 91 57)(54 109 75 92 58)(55 110 76 93 59)(56 111 77 94 60)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 65)(24 66)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 105)
(1 11 50)(2 51 12)(3 13 52)(4 53 14)(5 15 54)(6 55 16)(7 9 56)(8 49 10)(17 29 106)(18 107 30)(19 31 108)(20 109 32)(21 25 110)(22 111 26)(23 27 112)(24 105 28)(33 79 113)(34 114 80)(35 73 115)(36 116 74)(37 75 117)(38 118 76)(39 77 119)(40 120 78)(41 58 103)(42 104 59)(43 60 97)(44 98 61)(45 62 99)(46 100 63)(47 64 101)(48 102 57)(65 87 95)(66 96 88)(67 81 89)(68 90 82)(69 83 91)(70 92 84)(71 85 93)(72 94 86)
(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)

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

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

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

D5×C3⋊C8 is a maximal subgroup of
C30.C42  C30.3C42  C30.4C42  Dic5.Dic6  Dic5.4Dic6  D10.Dic6  D10.2Dic6  S3×C8×D5  C40.55D6  C40.35D6  D20.3Dic3  D20.2Dic3  D12.24D10  C60.16C23  D20.14D6  D20.D6
D5×C3⋊C8 is a maximal quotient of
C40.51D6  C60.93D4  C60.13Q8

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + + - image C1 C2 C2 C2 C4 C4 C8 S3 D5 Dic3 D6 Dic3 D10 C3⋊C8 C4×D5 C8×D5 S3×D5 D5×Dic3 D5×C3⋊C8 kernel D5×C3⋊C8 C5×C3⋊C8 C15⋊3C8 D5×C12 C3×Dic5 C6×D5 C3×D5 C4×D5 C3⋊C8 Dic5 C20 D10 C12 D5 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 2 8 1 2 1 1 1 2 4 4 8 2 2 4

Matrix representation of D5×C3⋊C8 in GL4(𝔽241) generated by

 240 1 0 0 188 52 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 188 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 240 0 0 1 240
,
 30 0 0 0 0 30 0 0 0 0 166 126 0 0 51 75
G:=sub<GL(4,GF(241))| [240,188,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[240,188,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[30,0,0,0,0,30,0,0,0,0,166,51,0,0,126,75] >;

D5×C3⋊C8 in GAP, Magma, Sage, TeX

D_5\times C_3\rtimes C_8
% in TeX

G:=Group("D5xC3:C8");
// GroupNames label

G:=SmallGroup(240,7);
// by ID

G=gap.SmallGroup(240,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,31,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^3=d^8=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^-1=c^-1>;
// generators/relations

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