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## G = D60⋊11C2order 240 = 24·3·5

### The semidirect product of D60 and C2 acting through Inn(D60)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60⋊11C2
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15 — D60⋊11C2
 Lower central C15 — C30 — D60⋊11C2
 Upper central C1 — C4 — C2×C4

Generators and relations for D6011C2
G = < a,b,c | a60=b2=c2=1, bab=a-1, ac=ca, cbc=a30b >

Subgroups: 392 in 80 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, D15, C30, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C4○D12, Dic15, C60, D30, C2×C30, C4○D20, Dic30, C4×D15, D60, C157D4, C2×C60, D6011C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C22×D5, C4○D12, D30, C4○D20, C22×D15, D6011C2

Smallest permutation representation of D6011C2
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)```

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D15 C4○D12 D30 D30 C4○D20 D60⋊11C2 kernel D60⋊11C2 Dic30 C4×D15 D60 C15⋊7D4 C2×C60 C2×C20 C2×C12 C20 C2×C10 C15 C12 C2×C6 C2×C4 C5 C4 C22 C3 C1 # reps 1 1 2 1 2 1 1 2 2 1 2 4 2 4 4 8 4 8 16

Matrix representation of D6011C2 in GL2(𝔽61) generated by

 49 55 6 8
,
 37 14 33 24
,
 14 45 16 47
`G:=sub<GL(2,GF(61))| [49,6,55,8],[37,33,14,24],[14,16,45,47] >;`

D6011C2 in GAP, Magma, Sage, TeX

`D_{60}\rtimes_{11}C_2`
`% in TeX`

`G:=Group("D60:11C2");`
`// GroupNames label`

`G:=SmallGroup(240,178);`
`// by ID`

`G=gap.SmallGroup(240,178);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,964,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^30*b>;`
`// generators/relations`

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