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G = D30⋊3C4order 240 = 24·3·5

1st semidirect product of D30 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D30⋊3C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — D30⋊3C4
 Lower central C15 — C30 — D30⋊3C4
 Upper central C1 — C22 — C2×C4

Generators and relations for D303C4
G = < a,b,c | a30=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >

Subgroups: 392 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C22⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×C12, C22×S3, D15, C30, C2×Dic5, C2×C20, C22×D5, D6⋊C4, Dic15, C60, D30, D30, C2×C30, D10⋊C4, C2×Dic15, C2×C60, C22×D15, D303C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C4×D15, D60, C157D4, D303C4

Smallest permutation representation of D303C4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 116 51 71)(2 117 52 72)(3 118 53 73)(4 119 54 74)(5 120 55 75)(6 91 56 76)(7 92 57 77)(8 93 58 78)(9 94 59 79)(10 95 60 80)(11 96 31 81)(12 97 32 82)(13 98 33 83)(14 99 34 84)(15 100 35 85)(16 101 36 86)(17 102 37 87)(18 103 38 88)(19 104 39 89)(20 105 40 90)(21 106 41 61)(22 107 42 62)(23 108 43 63)(24 109 44 64)(25 110 45 65)(26 111 46 66)(27 112 47 67)(28 113 48 68)(29 114 49 69)(30 115 50 70)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 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 2 3 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 1 1 30 30 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D5 D6 D10 C4×S3 D12 C3⋊D4 D15 C4×D5 D20 C5⋊D4 D30 C4×D15 D60 C15⋊7D4 kernel D30⋊3C4 C2×Dic15 C2×C60 C22×D15 D30 C2×C20 C30 C2×C12 C2×C10 C2×C6 C10 C10 C10 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 2 1 2 2 2 2 4 4 4 4 4 8 8 8

Matrix representation of D303C4 in GL4(𝔽61) generated by

 23 53 0 0 8 45 0 0 0 0 52 24 0 0 32 23
,
 23 53 0 0 5 38 0 0 0 0 45 31 0 0 39 16
,
 11 0 0 0 0 11 0 0 0 0 30 16 0 0 1 31
`G:=sub<GL(4,GF(61))| [23,8,0,0,53,45,0,0,0,0,52,32,0,0,24,23],[23,5,0,0,53,38,0,0,0,0,45,39,0,0,31,16],[11,0,0,0,0,11,0,0,0,0,30,1,0,0,16,31] >;`

D303C4 in GAP, Magma, Sage, TeX

`D_{30}\rtimes_3C_4`
`% in TeX`

`G:=Group("D30:3C4");`
`// GroupNames label`

`G:=SmallGroup(240,75);`
`// by ID`

`G=gap.SmallGroup(240,75);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,964,6917]);`
`// Polycyclic`

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

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