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## G = C30.38D4order 240 = 24·3·5

### 7th non-split extension by C30 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.38D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C30.38D4
 Lower central C15 — C30 — C30.38D4
 Upper central C1 — C22 — C23

Generators and relations for C30.38D4
G = < a,b,c | a30=b4=1, c2=a15, bab-1=cac-1=a-1, cbc-1=a15b-1 >

Subgroups: 232 in 68 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, Dic5, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C30, C30, C30, C2×Dic5, C22×C10, C6.D4, Dic15, C2×C30, C2×C30, C2×C30, C23.D5, C2×Dic15, C22×C30, C30.38D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, C2×Dic15, C157D4, C30.38D4

Smallest permutation representation of C30.38D4
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 35 86 105)(2 34 87 104)(3 33 88 103)(4 32 89 102)(5 31 90 101)(6 60 61 100)(7 59 62 99)(8 58 63 98)(9 57 64 97)(10 56 65 96)(11 55 66 95)(12 54 67 94)(13 53 68 93)(14 52 69 92)(15 51 70 91)(16 50 71 120)(17 49 72 119)(18 48 73 118)(19 47 74 117)(20 46 75 116)(21 45 76 115)(22 44 77 114)(23 43 78 113)(24 42 79 112)(25 41 80 111)(26 40 81 110)(27 39 82 109)(28 38 83 108)(29 37 84 107)(30 36 85 106)
(1 120 16 105)(2 119 17 104)(3 118 18 103)(4 117 19 102)(5 116 20 101)(6 115 21 100)(7 114 22 99)(8 113 23 98)(9 112 24 97)(10 111 25 96)(11 110 26 95)(12 109 27 94)(13 108 28 93)(14 107 29 92)(15 106 30 91)(31 90 46 75)(32 89 47 74)(33 88 48 73)(34 87 49 72)(35 86 50 71)(36 85 51 70)(37 84 52 69)(38 83 53 68)(39 82 54 67)(40 81 55 66)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A ··· 6G 10A ··· 10N 15A 15B 15C 15D 30A ··· 30AB order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 ··· 6 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 2 2 2 30 30 30 30 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2

66 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + + - + image C1 C2 C2 C4 S3 D4 D5 Dic3 D6 Dic5 D10 C3⋊D4 D15 C5⋊D4 Dic15 D30 C15⋊7D4 kernel C30.38D4 C2×Dic15 C22×C30 C2×C30 C22×C10 C30 C22×C6 C2×C10 C2×C10 C2×C6 C2×C6 C10 C23 C6 C22 C22 C2 # reps 1 2 1 4 1 2 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C30.38D4 in GL3(𝔽61) generated by

 60 0 0 0 49 0 0 0 5
,
 11 0 0 0 0 1 0 1 0
,
 50 0 0 0 0 1 0 60 0
`G:=sub<GL(3,GF(61))| [60,0,0,0,49,0,0,0,5],[11,0,0,0,0,1,0,1,0],[50,0,0,0,0,60,0,1,0] >;`

C30.38D4 in GAP, Magma, Sage, TeX

`C_{30}._{38}D_4`
`% in TeX`

`G:=Group("C30.38D4");`
`// GroupNames label`

`G:=SmallGroup(240,80);`
`// by ID`

`G=gap.SmallGroup(240,80);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,964,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^4=1,c^2=a^15,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^15*b^-1>;`
`// generators/relations`

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