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## G = D30.45D4order 480 = 25·3·5

### 18th non-split extension by D30 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D30.45D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×D30.C2 — D30.45D4
 Lower central C15 — C30 — D30.45D4
 Upper central C1 — C22 — C23

Generators and relations for D30.45D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=dbd-1=a18b, dcd-1=a15c-1 >

Subgroups: 1740 in 264 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, D15, C30, C30, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C5×Dic3, C3×Dic5, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, S3×C22⋊C4, D30.C2, C6×Dic5, C10×Dic3, C22×D15, C22×D15, C22×D15, C22×C30, D5×C22⋊C4, D304C4, C3×C23.D5, C5×C6.D4, C2×D30.C2, C23×D15, D30.45D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, C22×S3, C2×C22⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, S3×D5, C2×C4×D5, D4×D5, S3×C22⋊C4, D30.C2, C2×S3×D5, D5×C22⋊C4, C2×D30.C2, D10⋊D6, D30.45D4

Smallest permutation representation of D30.45D4
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 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 97 71 54)(2 116 72 43)(3 105 73 32)(4 94 74 51)(5 113 75 40)(6 102 76 59)(7 91 77 48)(8 110 78 37)(9 99 79 56)(10 118 80 45)(11 107 81 34)(12 96 82 53)(13 115 83 42)(14 104 84 31)(15 93 85 50)(16 112 86 39)(17 101 87 58)(18 120 88 47)(19 109 89 36)(20 98 90 55)(21 117 61 44)(22 106 62 33)(23 95 63 52)(24 114 64 41)(25 103 65 60)(26 92 66 49)(27 111 67 38)(28 100 68 57)(29 119 69 46)(30 108 70 35)
(1 112 16 97)(2 101 17 116)(3 120 18 105)(4 109 19 94)(5 98 20 113)(6 117 21 102)(7 106 22 91)(8 95 23 110)(9 114 24 99)(10 103 25 118)(11 92 26 107)(12 111 27 96)(13 100 28 115)(14 119 29 104)(15 108 30 93)(31 84 46 69)(32 73 47 88)(33 62 48 77)(34 81 49 66)(35 70 50 85)(36 89 51 74)(37 78 52 63)(38 67 53 82)(39 86 54 71)(40 75 55 90)(41 64 56 79)(42 83 57 68)(43 72 58 87)(44 61 59 76)(45 80 60 65)```

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 D10 D10 C4×S3 C4×D5 S3×D4 S3×D5 D4×D5 D30.C2 C2×S3×D5 D10⋊D6 kernel D30.45D4 D30⋊4C4 C3×C23.D5 C5×C6.D4 C2×D30.C2 C23×D15 C22×D15 C23.D5 D30 C6.D4 C2×Dic5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C6 C10 C23 C6 C22 C22 C2 # reps 1 2 1 1 2 1 8 1 4 2 2 1 4 2 4 8 2 2 4 4 2 8

Matrix representation of D30.45D4 in GL6(𝔽61)

 43 60 0 0 0 0 1 0 0 0 0 0 0 0 60 56 0 0 0 0 25 2 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 43 60 0 0 0 0 18 18 0 0 0 0 0 0 60 56 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 43 60 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 50 52 0 0 0 0 54 11
,
 60 0 0 0 0 0 18 1 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 50 52 0 0 0 0 0 11

`G:=sub<GL(6,GF(61))| [43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,25,0,0,0,0,56,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,56,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,54,0,0,0,0,52,11],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,52,11] >;`

D30.45D4 in GAP, Magma, Sage, TeX

`D_{30}._{45}D_4`
`% in TeX`

`G:=Group("D30.45D4");`
`// GroupNames label`

`G:=SmallGroup(480,637);`
`// by ID`

`G=gap.SmallGroup(480,637);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=d*b*d^-1=a^18*b,d*c*d^-1=a^15*c^-1>;`
`// generators/relations`

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