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## G = D60⋊C22order 480 = 25·3·5

### 6th semidirect product of D60 and C22 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D60⋊C22
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C20⋊D6 — D60⋊C22
 Lower central C15 — C30 — C60 — D60⋊C22
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D60⋊C22
G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a11, dad=a29, cbc=a55b, dbd=a58b, cd=dc >

Subgroups: 1020 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3 [×2], D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C52C8, C40, C4×D5 [×2], D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, Dic15, C60, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, Q83D6, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, C4×D15, C4×D15, D60, D60, Q8×C15, C2×S3×D5, D40⋊C2, D30.5C4, C3⋊D40, C5⋊D24, C3×Q8⋊D5, C5×Q82S3, C20⋊D6, Q83D15, D60⋊C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q83D6, C2×S3×D5, D40⋊C2, D10⋊D6, D60⋊C22

Smallest permutation representation of D60⋊C22
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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)
(2 12)(3 23)(4 34)(5 45)(6 56)(8 18)(9 29)(10 40)(11 51)(14 24)(15 35)(16 46)(17 57)(20 30)(21 41)(22 52)(26 36)(27 47)(28 58)(32 42)(33 53)(38 48)(39 59)(44 54)(50 60)(61 96)(62 107)(63 118)(64 69)(65 80)(66 91)(67 102)(68 113)(70 75)(71 86)(72 97)(73 108)(74 119)(76 81)(77 92)(78 103)(79 114)(82 87)(83 98)(84 109)(85 120)(88 93)(89 104)(90 115)(94 99)(95 110)(100 105)(101 116)(106 111)(112 117)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 75)(62 104)(63 73)(64 102)(65 71)(66 100)(67 69)(68 98)(70 96)(72 94)(74 92)(76 90)(77 119)(78 88)(79 117)(80 86)(81 115)(82 84)(83 113)(85 111)(87 109)(89 107)(91 105)(93 103)(95 101)(97 99)(106 120)(108 118)(110 116)(112 114)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C8⋊C22 S3×D4 S3×D5 D4×D5 Q8⋊3D6 C2×S3×D5 D40⋊C2 D10⋊D6 D60⋊C22 kernel D60⋊C22 D30.5C4 C3⋊D40 C5⋊D24 C3×Q8⋊D5 C5×Q8⋊2S3 C20⋊D6 Q8⋊3D15 Q8⋊D5 Dic15 D30 Q8⋊2S3 C5⋊2C8 D20 C5×Q8 C3⋊C8 D12 C3×Q8 C15 C10 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 1 1 2 2 2 2 4 4 2

Matrix representation of D60⋊C22 in GL8(𝔽241)

 240 189 240 189 0 0 0 0 52 52 52 52 0 0 0 0 1 52 0 0 0 0 0 0 189 189 0 0 0 0 0 0 0 0 0 0 1 190 2 139 0 0 0 0 51 51 102 102 0 0 0 0 240 51 240 51 0 0 0 0 190 190 190 190
,
 38 28 203 213 0 0 0 0 221 203 20 38 0 0 0 0 165 185 203 213 0 0 0 0 40 76 20 38 0 0 0 0 0 0 0 0 0 0 237 29 0 0 0 0 0 0 66 4 0 0 0 0 239 135 0 0 0 0 0 0 33 2 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 240 0 0 0 0 0 0 240 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 240 0 0 0 0 0 0 240 0 240
,
 1 0 0 0 0 0 0 0 189 240 0 0 0 0 0 0 240 0 240 0 0 0 0 0 52 1 52 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 51 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 51 240

`G:=sub<GL(8,GF(241))| [240,52,1,189,0,0,0,0,189,52,52,189,0,0,0,0,240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,0,0,1,51,240,190,0,0,0,0,190,51,51,190,0,0,0,0,2,102,240,190,0,0,0,0,139,102,51,190],[38,221,165,40,0,0,0,0,28,203,185,76,0,0,0,0,203,20,203,20,0,0,0,0,213,38,213,38,0,0,0,0,0,0,0,0,0,0,239,33,0,0,0,0,0,0,135,2,0,0,0,0,237,66,0,0,0,0,0,0,29,4,0,0],[1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,189,240,52,0,0,0,0,0,240,0,1,0,0,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240] >;`

D60⋊C22 in GAP, Magma, Sage, TeX

`D_{60}\rtimes C_2^2`
`% in TeX`

`G:=Group("D60:C2^2");`
`// GroupNames label`

`G:=SmallGroup(480,582);`
`// by ID`

`G=gap.SmallGroup(480,582);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^60=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^11,d*a*d=a^29,c*b*c=a^55*b,d*b*d=a^58*b,c*d=d*c>;`
`// generators/relations`

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