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## G = C3×D5.D8order 480 = 25·3·5

### Direct product of C3 and D5.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D5.D8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4⋊F5 — C3×D5.D8
 Lower central C5 — C10 — C20 — C3×D5.D8
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×D5.D8
G = < a,b,c,d,e | a3=b5=c2=d8=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=d-1 >

Subgroups: 280 in 72 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C2.D8, C52C8, C40, C4×D5, C2×F5, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, C4⋊F5, C3×C2.D8, C3×C52C8, C120, D5×C12, C6×F5, D5.D8, D5×C24, C3×C4⋊F5, C3×D5.D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, D8, Q16, F5, C2×C12, C3×D4, C3×Q8, C2.D8, C2×F5, C3×C4⋊C4, C3×D8, C3×Q16, C3×F5, C4⋊F5, C3×C2.D8, C6×F5, D5.D8, C3×C4⋊F5, C3×D5.D8

Smallest permutation representation of C3×D5.D8
On 120 points
Generators in S120
(1 104 79)(2 97 80)(3 98 73)(4 99 74)(5 100 75)(6 101 76)(7 102 77)(8 103 78)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 89 114)(18 90 115)(19 91 116)(20 92 117)(21 93 118)(22 94 119)(23 95 120)(24 96 113)(25 45 61)(26 46 62)(27 47 63)(28 48 64)(29 41 57)(30 42 58)(31 43 59)(32 44 60)(33 108 69)(34 109 70)(35 110 71)(36 111 72)(37 112 65)(38 105 66)(39 106 67)(40 107 68)
(1 49 20 46 105)(2 50 21 47 106)(3 51 22 48 107)(4 52 23 41 108)(5 53 24 42 109)(6 54 17 43 110)(7 55 18 44 111)(8 56 19 45 112)(9 94 64 68 98)(10 95 57 69 99)(11 96 58 70 100)(12 89 59 71 101)(13 90 60 72 102)(14 91 61 65 103)(15 92 62 66 104)(16 93 63 67 97)(25 37 78 83 116)(26 38 79 84 117)(27 39 80 85 118)(28 40 73 86 119)(29 33 74 87 120)(30 34 75 88 113)(31 35 76 81 114)(32 36 77 82 115)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)(65 103)(66 104)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)
(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 7)(2 6)(3 5)(9 96 68 58)(10 95 69 57)(11 94 70 64)(12 93 71 63)(13 92 72 62)(14 91 65 61)(15 90 66 60)(16 89 67 59)(17 106 43 50)(18 105 44 49)(19 112 45 56)(20 111 46 55)(21 110 47 54)(22 109 48 53)(23 108 41 52)(24 107 42 51)(25 83 116 37)(26 82 117 36)(27 81 118 35)(28 88 119 34)(29 87 120 33)(30 86 113 40)(31 85 114 39)(32 84 115 38)(73 75)(76 80)(77 79)(97 101)(98 100)(102 104)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + - + + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 Q8 D4 D8 Q16 C3×Q8 C3×D4 C3×D8 C3×Q16 F5 C2×F5 C3×F5 C4⋊F5 C6×F5 D5.D8 C3×C4⋊F5 C3×D5.D8 kernel C3×D5.D8 D5×C24 C3×C4⋊F5 D5.D8 C3×C5⋊2C8 C120 C8×D5 C4⋊F5 C5⋊2C8 C40 C3×Dic5 C6×D5 C3×D5 C3×D5 Dic5 D10 D5 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 2 2 2 2 4 4 1 1 2 2 2 4 4 8

Matrix representation of C3×D5.D8 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 1 3 3 0 1 6 3 4 4 2 2 1 5 6 0
,
 6 3 4 2 6 5 1 1 1 1 5 6 3 6 1 5
,
 6 3 0 2 1 6 1 1 2 2 2 3 3 4 2 6
,
 4 0 0 5 6 1 0 3 5 3 5 0 4 0 2 4
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,6,1,3,3,5,1,6,4,1,5,1,2,1,6,5],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[4,6,5,4,0,1,3,0,0,0,5,2,5,3,0,4] >;

C3×D5.D8 in GAP, Magma, Sage, TeX

C_3\times D_5.D_8
% in TeX

G:=Group("C3xD5.D8");
// GroupNames label

G:=SmallGroup(480,274);
// by ID

G=gap.SmallGroup(480,274);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,512,2524,102,9414,1595]);
// Polycyclic

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

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