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## G = C15⋊2+ 1+4order 480 = 25·3·5

### The semidirect product of C15 and 2+ 1+4 acting via 2+ 1+4/C23=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C15⋊2+ 1+4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×C3⋊D4 — C15⋊2+ 1+4
 Lower central C15 — C30 — C15⋊2+ 1+4
 Upper central C1 — C2 — C23

Generators and relations for C15⋊2+ 1+4
G = < a,b,c,d,e | a15=b4=c2=e2=1, d2=b2, bab-1=cac=a11, dad-1=a4, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1788 in 332 conjugacy classes, 108 normal (40 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C2×C30, C4○D20, D4×D5, D42D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D46D6, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, C157D4, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, D46D10, Dic5.D6, C30.C23, Dic3.D10, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, C6×C5⋊D4, C10×C3⋊D4, C2×C157D4, C15⋊2+ 1+4
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D46D6, C2×S3×D5, D46D10, C22×S3×D5, C15⋊2+ 1+4

Smallest permutation representation of C15⋊2+ 1+4
On 120 points
Generators in S120
```(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 81 24 63)(2 77 25 74)(3 88 26 70)(4 84 27 66)(5 80 28 62)(6 76 29 73)(7 87 30 69)(8 83 16 65)(9 79 17 61)(10 90 18 72)(11 86 19 68)(12 82 20 64)(13 78 21 75)(14 89 22 71)(15 85 23 67)(31 111 53 97)(32 107 54 93)(33 118 55 104)(34 114 56 100)(35 110 57 96)(36 106 58 92)(37 117 59 103)(38 113 60 99)(39 109 46 95)(40 120 47 91)(41 116 48 102)(42 112 49 98)(43 108 50 94)(44 119 51 105)(45 115 52 101)
(1 63)(2 74)(3 70)(4 66)(5 62)(6 73)(7 69)(8 65)(9 61)(10 72)(11 68)(12 64)(13 75)(14 71)(15 67)(16 83)(17 79)(18 90)(19 86)(20 82)(21 78)(22 89)(23 85)(24 81)(25 77)(26 88)(27 84)(28 80)(29 76)(30 87)(31 97)(32 93)(33 104)(34 100)(35 96)(36 92)(37 103)(38 99)(39 95)(40 91)(41 102)(42 98)(43 94)(44 105)(45 101)(46 109)(47 120)(48 116)(49 112)(50 108)(51 119)(52 115)(53 111)(54 107)(55 118)(56 114)(57 110)(58 106)(59 117)(60 113)
(1 54 24 32)(2 58 25 36)(3 47 26 40)(4 51 27 44)(5 55 28 33)(6 59 29 37)(7 48 30 41)(8 52 16 45)(9 56 17 34)(10 60 18 38)(11 49 19 42)(12 53 20 31)(13 57 21 35)(14 46 22 39)(15 50 23 43)(61 114 79 100)(62 118 80 104)(63 107 81 93)(64 111 82 97)(65 115 83 101)(66 119 84 105)(67 108 85 94)(68 112 86 98)(69 116 87 102)(70 120 88 91)(71 109 89 95)(72 113 90 99)(73 117 76 103)(74 106 77 92)(75 110 78 96)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 31)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 117)(77 118)(78 119)(79 120)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)```

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 30 ··· 30 size 1 1 2 2 2 6 6 10 10 30 30 2 6 6 10 10 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 4 4 4 4 12 12 12 12 20 20 4 4 12 12 12 12 4 ··· 4

63 irreducible representations

 dim 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 C2 C2 S3 D5 D6 D6 D6 D6 D10 D10 D10 D10 2+ 1+4 S3×D5 D4⋊6D6 C2×S3×D5 D4⋊6D10 C15⋊2+ 1+4 kernel C15⋊2+ 1+4 Dic5.D6 C30.C23 Dic3.D10 D5×C3⋊D4 S3×C5⋊D4 D10⋊D6 C6×C5⋊D4 C10×C3⋊D4 C2×C15⋊7D4 C2×C5⋊D4 C2×C3⋊D4 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C15 C23 C5 C22 C3 C1 # reps 1 2 2 2 2 2 2 1 1 1 1 2 1 4 1 1 2 8 2 2 1 2 2 6 4 8

Matrix representation of C15⋊2+ 1+4 in GL4(𝔽61) generated by

 8 6 0 0 8 0 0 0 53 56 48 38 44 35 23 23
,
 0 0 60 1 11 46 59 17 4 50 15 0 3 50 15 0
,
 0 0 60 1 11 46 59 17 18 42 15 0 19 42 15 0
,
 8 14 0 0 52 53 0 0 4 0 14 45 36 34 39 47
,
 30 45 0 0 60 31 0 0 4 0 14 45 11 57 16 47
`G:=sub<GL(4,GF(61))| [8,8,53,44,6,0,56,35,0,0,48,23,0,0,38,23],[0,11,4,3,0,46,50,50,60,59,15,15,1,17,0,0],[0,11,18,19,0,46,42,42,60,59,15,15,1,17,0,0],[8,52,4,36,14,53,0,34,0,0,14,39,0,0,45,47],[30,60,4,11,45,31,0,57,0,0,14,16,0,0,45,47] >;`

C15⋊2+ 1+4 in GAP, Magma, Sage, TeX

`C_{15}\rtimes 2_+^{1+4}`
`% in TeX`

`G:=Group("C15:ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(480,1125);`
`// by ID`

`G=gap.SmallGroup(480,1125);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^15=b^4=c^2=e^2=1,d^2=b^2,b*a*b^-1=c*a*c=a^11,d*a*d^-1=a^4,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;`
`// generators/relations`

׿
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