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## G = C60.97D4order 480 = 25·3·5

### 97th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.97D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C3×C4○D20 — C60.97D4
 Lower central C15 — C30 — C60 — C60.97D4
 Upper central C1 — C4 — C2×C4

Generators and relations for C60.97D4
G = < a,b,c | a60=b4=1, c2=a45, bab-1=a41, cac-1=a29, cbc-1=a45b-1 >

Subgroups: 364 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C6×D5, C2×C30, C4.Dic5, C4×C20, C4○D20, Q83Dic3, C153C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C10×Dic3, C2×C60, D204C4, Dic3×C20, C60.7C4, C3×C4○D20, C60.97D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, Q83Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D204C4, D10⋊Dic3, C60.97D4

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

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - - + + + + - + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D5 Dic3 Dic3 D6 D10 C3⋊D4 C3⋊D4 C4≀C2 C4×D5 D20 C5⋊D4 D20⋊4C4 S3×D5 Q8⋊3Dic3 D5×Dic3 C3⋊D20 C15⋊D4 C60.97D4 kernel C60.97D4 Dic3×C20 C60.7C4 C3×C4○D20 C3×Dic10 C3×D20 C4○D20 C60 C2×C30 C4×Dic3 Dic10 D20 C2×C20 C2×C12 C20 C2×C10 C15 C12 C12 C2×C6 C3 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 1 2 1 1 1 2 2 2 4 4 4 4 16 2 2 2 2 2 8

Matrix representation of C60.97D4 in GL4(𝔽241) generated by

 240 1 0 0 240 0 0 0 0 0 135 0 0 0 0 216
,
 0 1 0 0 1 0 0 0 0 0 240 0 0 0 0 64
,
 0 240 0 0 240 0 0 0 0 0 0 64 0 0 240 0
`G:=sub<GL(4,GF(241))| [240,240,0,0,1,0,0,0,0,0,135,0,0,0,0,216],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,64],[0,240,0,0,240,0,0,0,0,0,0,240,0,0,64,0] >;`

C60.97D4 in GAP, Magma, Sage, TeX

`C_{60}._{97}D_4`
`% in TeX`

`G:=Group("C60.97D4");`
`// GroupNames label`

`G:=SmallGroup(480,53);`
`// by ID`

`G=gap.SmallGroup(480,53);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^45,b*a*b^-1=a^41,c*a*c^-1=a^29,c*b*c^-1=a^45*b^-1>;`
`// generators/relations`

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