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## G = D60.C22order 480 = 25·3·5

### 2nd non-split extension by D60 of 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 — S3×D20 — D60.C22
 Lower central C15 — C30 — C60 — D60.C22
 Upper central C1 — C2 — C4 — D4

Generators and relations for D60.C22
G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a19, dad=a31, cbc=a18b, dbd=a45b, dcd=a45c >

Subgroups: 876 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C52C8, D20, D20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, C5×Dic3, C60, S3×D5, C6×D5, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, D8⋊S3, C3×C52C8, C153C8, C3⋊D20, C3×D20, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D60, D4×C15, C2×S3×D5, D4⋊D10, D6.Dic5, C30.D4, Dic6⋊D5, C3×D4⋊D5, D4⋊D15, S3×D20, C5×D42S3, D60.C22
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊S3, C2×S3×D5, D4⋊D10, S3×C5⋊D4, 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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(118 120)
(2 20)(3 39)(4 58)(5 17)(6 36)(7 55)(8 14)(9 33)(10 52)(12 30)(13 49)(15 27)(16 46)(18 24)(19 43)(22 40)(23 59)(25 37)(26 56)(28 34)(29 53)(32 50)(35 47)(38 44)(42 60)(45 57)(48 54)(61 88)(62 107)(63 66)(64 85)(65 104)(67 82)(68 101)(69 120)(70 79)(71 98)(72 117)(73 76)(74 95)(75 114)(77 92)(78 111)(80 89)(81 108)(83 86)(84 105)(87 102)(90 99)(91 118)(93 96)(94 115)(97 112)(100 109)(103 106)(110 119)(113 116)
(1 112)(2 83)(3 114)(4 85)(5 116)(6 87)(7 118)(8 89)(9 120)(10 91)(11 62)(12 93)(13 64)(14 95)(15 66)(16 97)(17 68)(18 99)(19 70)(20 101)(21 72)(22 103)(23 74)(24 105)(25 76)(26 107)(27 78)(28 109)(29 80)(30 111)(31 82)(32 113)(33 84)(34 115)(35 86)(36 117)(37 88)(38 119)(39 90)(40 61)(41 92)(42 63)(43 94)(44 65)(45 96)(46 67)(47 98)(48 69)(49 100)(50 71)(51 102)(52 73)(53 104)(54 75)(55 106)(56 77)(57 108)(58 79)(59 110)(60 81)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 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 C5⋊D4 C5⋊D4 C8⋊C22 S3×D4 S3×D5 D8⋊S3 C2×S3×D5 D4⋊D10 S3×C5⋊D4 D60.C22 kernel D60.C22 D6.Dic5 C30.D4 Dic6⋊D5 C3×D4⋊D5 D4⋊D15 S3×D20 C5×D4⋊2S3 D4⋊D5 C5×Dic3 S3×C10 D4⋊2S3 C5⋊2C8 D20 C5×D4 Dic6 C4×S3 C3×D4 Dic3 D6 C15 C10 D4 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 4 4 1 1 2 2 2 4 4 2

Matrix representation of D60.C22 in GL6(𝔽241)

 189 51 0 0 0 0 189 0 0 0 0 0 0 0 0 0 240 240 0 0 0 0 1 0 0 0 1 1 0 0 0 0 240 0 0 0
,
 240 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 240 0 0 0 0 0 1 0 0 240 240 0 0 0 0 0 1 0 0
,
 189 1 0 0 0 0 189 52 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 47 94 47 94 0 0 147 194 147 194 0 0 47 94 194 147 0 0 147 194 94 47

`G:=sub<GL(6,GF(241))| [189,189,0,0,0,0,51,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,240,1,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,240,1,0,0,240,0,0,0,0,0,240,1,0,0],[189,189,0,0,0,0,1,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,94,194,94,194,0,0,47,147,194,94,0,0,94,194,147,47] >;`

D60.C22 in GAP, Magma, Sage, TeX

`D_{60}.C_2^2`
`% in TeX`

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

`G:=SmallGroup(480,556);`
`// by ID`

`G=gap.SmallGroup(480,556);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,1356,18822]);`
`// 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^19,d*a*d=a^31,c*b*c=a^18*b,d*b*d=a^45*b,d*c*d=a^45*c>;`
`// generators/relations`

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