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## G = C60.98D4order 480 = 25·3·5

### 98th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.98D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.98D4
 Lower central C15 — C30 — C60 — C60.98D4
 Upper central C1 — C4 — C2×C4

Generators and relations for C60.98D4
G = < a,b,c,d | a12=c10=1, b2=a6, d2=c5, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 380 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C10, C10 [×2], Dic3 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, Dic15 [×2], C60 [×2], S3×C10, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, D12⋊C4, C3×C52C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×Dic15, C2×C60, D42Dic5, C3×C4.Dic5, C4×Dic15, C5×C4○D12, C60.98D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4≀C2, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, D12⋊C4, S3×Dic5, C15⋊D4, C5⋊D12, D42Dic5, D6⋊Dic5, C60.98D4

Smallest permutation representation of C60.98D4
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 32 7 26)(2 31 8 25)(3 30 9 36)(4 29 10 35)(5 28 11 34)(6 27 12 33)(13 54 19 60)(14 53 20 59)(15 52 21 58)(16 51 22 57)(17 50 23 56)(18 49 24 55)(37 75 43 81)(38 74 44 80)(39 73 45 79)(40 84 46 78)(41 83 47 77)(42 82 48 76)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)(97 116 103 110)(98 115 104 109)(99 114 105 120)(100 113 106 119)(101 112 107 118)(102 111 108 117)
(1 104 24 63 79)(2 105 13 64 80)(3 106 14 65 81)(4 107 15 66 82)(5 108 16 67 83)(6 97 17 68 84)(7 98 18 69 73)(8 99 19 70 74)(9 100 20 71 75)(10 101 21 72 76)(11 102 22 61 77)(12 103 23 62 78)(25 120 60 88 44 31 114 54 94 38)(26 109 49 89 45 32 115 55 95 39)(27 110 50 90 46 33 116 56 96 40)(28 111 51 91 47 34 117 57 85 41)(29 112 52 92 48 35 118 58 86 42)(30 113 53 93 37 36 119 59 87 43)
(1 7)(2 12)(3 5)(4 10)(6 8)(9 11)(13 62)(14 67)(15 72)(16 65)(17 70)(18 63)(19 68)(20 61)(21 66)(22 71)(23 64)(24 69)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 120 43 114)(38 113 44 119)(39 118 45 112)(40 111 46 117)(41 116 47 110)(42 109 48 115)(49 92 55 86)(50 85 56 91)(51 90 57 96)(52 95 58 89)(53 88 59 94)(54 93 60 87)(73 104)(74 97)(75 102)(76 107)(77 100)(78 105)(79 98)(80 103)(81 108)(82 101)(83 106)(84 99)```

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + - - + + + - - + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D5 D6 Dic5 Dic5 D10 C4×S3 C3⋊D4 D12 C4≀C2 C5⋊D4 C5⋊D4 S3×D5 D12⋊C4 S3×Dic5 C15⋊D4 C5⋊D12 D4⋊2Dic5 C60.98D4 kernel C60.98D4 C3×C4.Dic5 C4×Dic15 C5×C4○D12 C5×Dic6 C5×D12 C4.Dic5 C60 C2×C30 C4○D12 C2×C20 Dic6 D12 C2×C12 C20 C20 C2×C10 C15 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 1 1 1 2 1 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 8

Matrix representation of C60.98D4 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 64 0 0 0 0 0 0 177 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 76 89 0 0 0 0 103 165 0 0 0 0 0 0 0 177 0 0 0 0 177 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 240 191 0 0 0 0 240 190 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 51 191 0 0 0 0 52 190 0 0 0 0 0 0 240 0 0 0 0 0 0 177 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[76,103,0,0,0,0,89,165,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[240,240,0,0,0,0,191,190,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,52,0,0,0,0,191,190,0,0,0,0,0,0,240,0,0,0,0,0,0,177,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C60.98D4 in GAP, Magma, Sage, TeX

`C_{60}._{98}D_4`
`% in TeX`

`G:=Group("C60.98D4");`
`// GroupNames label`

`G:=SmallGroup(480,54);`
`// by ID`

`G=gap.SmallGroup(480,54);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=c^10=1,b^2=a^6,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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