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## G = C3×D8⋊D5order 480 = 25·3·5

### Direct product of C3 and D8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D8⋊D5
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×D4×D5 — C3×D8⋊D5
 Lower central C5 — C10 — C20 — C3×D8⋊D5
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×D8⋊D5
G = < a,b,c,d,e | a3=b8=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 544 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), D8, D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C3×M4(2), C3×D8, C3×D8, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, C3×C8⋊C22, C3×C52C8, C120, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, D4×C15, D5×C2×C6, D8⋊D5, C3×C8⋊D5, C3×C40⋊C2, C3×D4⋊D5, C3×D4.D5, C15×D8, C3×D4×D5, C3×D42D5, C3×D8⋊D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8⋊C22, D5×C2×C6, D8⋊D5, C3×D4×D5, C3×D8⋊D5

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

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

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

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

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C6×D5 C6×D5 C8⋊C22 D4×D5 C3×C8⋊C22 D8⋊D5 C3×D4×D5 C3×D8⋊D5 kernel C3×D8⋊D5 C3×C8⋊D5 C3×C40⋊C2 C3×D4⋊D5 C3×D4.D5 C15×D8 C3×D4×D5 C3×D4⋊2D5 D8⋊D5 C8⋊D5 C40⋊C2 D4⋊D5 D4.D5 C5×D8 D4×D5 D4⋊2D5 C3×Dic5 C6×D5 C3×D8 C24 C3×D4 Dic5 D10 D8 C8 D4 C15 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 4 2 2 4 4 8 1 2 2 4 4 8

Matrix representation of C3×D8⋊D5 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 0 0 173 151 0 0 173 22 55 225 15 144 170 71 211 226
,
 1 0 7 7 0 1 234 0 0 0 240 0 0 0 0 240
,
 189 240 0 0 1 0 0 0 0 0 240 240 0 0 191 190
,
 189 240 0 0 52 52 0 0 0 0 0 189 0 0 190 0
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,0,55,170,0,0,225,71,173,173,15,211,151,22,144,226],[1,0,0,0,0,1,0,0,7,234,240,0,7,0,0,240],[189,1,0,0,240,0,0,0,0,0,240,191,0,0,240,190],[189,52,0,0,240,52,0,0,0,0,0,190,0,0,189,0] >;

C3×D8⋊D5 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes D_5
% in TeX

G:=Group("C3xD8:D5");
// GroupNames label

G:=SmallGroup(480,704);
// by ID

G=gap.SmallGroup(480,704);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,303,1271,648,102,18822]);
// Polycyclic

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

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