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## G = C3×C8⋊D5order 240 = 24·3·5

### Direct product of C3 and C8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C8⋊D5
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×C8⋊D5
 Lower central C5 — C10 — C3×C8⋊D5
 Upper central C1 — C12 — C24

Generators and relations for C3×C8⋊D5
G = < a,b,c,d | a3=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

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

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

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

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

C3×C8⋊D5 is a maximal subgroup of
C40⋊D6  C24⋊D10  D24⋊D5  Dic60⋊C2  C24.2D10  C40.55D6  C40.35D6  C3×D5×M4(2)

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D5 M4(2) D10 C3×D5 C4×D5 C3×M4(2) C6×D5 C8⋊D5 D5×C12 C3×C8⋊D5 kernel C3×C8⋊D5 C3×C5⋊2C8 C120 D5×C12 C8⋊D5 C3×Dic5 C6×D5 C5⋊2C8 C40 C4×D5 Dic5 D10 C24 C15 C12 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 2 2 2 4 4 4 4 8 8 16

Matrix representation of C3×C8⋊D5 in GL3(𝔽241) generated by

 225 0 0 0 1 0 0 0 1
,
 1 0 0 0 216 140 0 101 25
,
 1 0 0 0 0 240 0 1 189
,
 240 0 0 0 52 240 0 52 189
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[1,0,0,0,216,101,0,140,25],[1,0,0,0,0,1,0,240,189],[240,0,0,0,52,52,0,240,189] >;

C3×C8⋊D5 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_5
% in TeX

G:=Group("C3xC8:D5");
// GroupNames label

G:=SmallGroup(240,34);
// by ID

G=gap.SmallGroup(240,34);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,69,6917]);
// Polycyclic

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

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