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## G = D30.C23order 480 = 25·3·5

### 13rd non-split extension by D30 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D30.C23
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — D30.C23
 Lower central C15 — C30 — D30.C23
 Upper central C1 — C2 — D4

Generators and relations for D30.C23
G = < a,b,c,d,e | a30=b2=c2=e2=1, d2=a15, bab=eae=a-1, cac=a11, ad=da, cbc=a25b, dbd-1=a15b, ebe=a28b, cd=dc, ce=ec, de=ed >

Subgroups: 1692 in 328 conjugacy classes, 110 normal (50 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×7], C22 [×2], C22 [×11], C5, S3 [×5], C6, C6 [×3], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3, Dic3 [×2], Dic3, C12, C12 [×3], D6, D6 [×9], C2×C6 [×2], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×2], Dic5, C20, C20 [×3], D10, D10 [×9], C2×C10 [×2], C2×C10, Dic6, Dic6 [×2], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×3], C5×S3, C3×D5, D15 [×2], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10, Dic10 [×2], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3, D42S3 [×2], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30 [×2], D30 [×4], C2×C30 [×2], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D42D5, D42D5 [×2], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, D30.C2 [×6], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], C4×D15, D60, C157D4 [×2], D4×C15, C2×S3×D5, C22×D15 [×2], D5×C4○D4, D60⋊C2, D15⋊Q8, C12.28D10, C4×S3×D5, Dic5.D6 [×2], Dic3.D10 [×2], C2×D30.C2 [×2], D10⋊D6 [×2], C3×D42D5, C5×D42S3, D4×D15, D30.C23
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D30.C23

Smallest permutation representation of D30.C23
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(61 95)(62 94)(63 93)(64 92)(65 91)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 36)(32 47)(33 58)(34 39)(35 50)(37 42)(38 53)(40 45)(41 56)(43 48)(44 59)(46 51)(49 54)(52 57)(55 60)(61 81)(63 73)(64 84)(66 76)(67 87)(69 79)(70 90)(72 82)(75 85)(78 88)(91 96)(92 107)(93 118)(94 99)(95 110)(97 102)(98 113)(100 105)(101 116)(103 108)(104 119)(106 111)(109 114)(112 117)(115 120)
(1 86 16 71)(2 87 17 72)(3 88 18 73)(4 89 19 74)(5 90 20 75)(6 61 21 76)(7 62 22 77)(8 63 23 78)(9 64 24 79)(10 65 25 80)(11 66 26 81)(12 67 27 82)(13 68 28 83)(14 69 29 84)(15 70 30 85)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 36)(32 35)(33 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(61 66)(62 65)(63 64)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(91 96)(92 95)(93 94)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)```

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 D10 S3×D5 S3×C4○D4 C2×S3×D5 C2×S3×D5 D5×C4○D4 D30.C23 kernel D30.C23 D60⋊C2 D15⋊Q8 C12.28D10 C4×S3×D5 Dic5.D6 Dic3.D10 C2×D30.C2 D10⋊D6 C3×D4⋊2D5 C5×D4⋊2S3 D4×D15 D4⋊2D5 D4⋊2S3 Dic10 C4×D5 C2×Dic5 C5⋊D4 C5×D4 D15 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 D4 C5 C4 C22 C3 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 2 1 1 2 2 1 4 2 2 4 4 2 2 2 2 4 4 2

Matrix representation of D30.C23 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 43 60 0 0 0 0 1 0 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 60 46 0 0 0 0 0 1 0 0 0 0 0 0 43 60 0 0 0 0 18 18 0 0 0 0 0 0 60 0 0 0 0 0 60 1
,
 1 0 0 0 0 0 8 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 50 0 0 0 0 0 34 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 43 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,46,1,0,0,0,0,0,0,43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[1,8,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,34,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D30.C23 in GAP, Magma, Sage, TeX

`D_{30}.C_2^3`
`% in TeX`

`G:=Group("D30.C2^3");`
`// GroupNames label`

`G:=SmallGroup(480,1100);`
`// by ID`

`G=gap.SmallGroup(480,1100);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,185,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^30=b^2=c^2=e^2=1,d^2=a^15,b*a*b=e*a*e=a^-1,c*a*c=a^11,a*d=d*a,c*b*c=a^25*b,d*b*d^-1=a^15*b,e*b*e=a^28*b,c*d=d*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

׿
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