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## G = C30.C23order 240 = 24·3·5

### 17th non-split extension by C30 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.C23
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C30.C23
 Lower central C15 — C30 — C30.C23
 Upper central C1 — C2 — C22

Generators and relations for C30.C23
G = < a,b,c,d | a30=b2=d2=1, c2=a15, bab=a19, cac-1=a11, ad=da, bc=cb, dbd=a15b, dcd=a15c >

Subgroups: 312 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, C30, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5⋊D4, C5×D4, D42S3, C5×Dic3, C3×Dic5, Dic15, C6×D5, S3×C10, C2×C30, D42D5, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C3×C5⋊D4, C5×C3⋊D4, C2×Dic15, C30.C23
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D42S3, S3×D5, D42D5, C2×S3×D5, C30.C23

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12 15A 15B 20A 20B 30A ··· 30F order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 12 15 15 20 20 30 ··· 30 size 1 1 2 6 10 2 6 10 15 15 30 2 2 2 4 20 2 2 4 4 12 12 20 4 4 12 12 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D4⋊2S3 S3×D5 D4⋊2D5 C2×S3×D5 C30.C23 kernel C30.C23 D5×Dic3 S3×Dic5 C15⋊D4 C15⋊Q8 C3×C5⋊D4 C5×C3⋊D4 C2×Dic15 C5⋊D4 C3⋊D4 Dic5 D10 C2×C10 C15 Dic3 D6 C2×C6 C5 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 1 2 2 2 4

Matrix representation of C30.C23 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 44 1 0 0 0 0 16 60 0 0 0 0 0 0 0 60 0 0 0 0 1 60
,
 1 0 0 0 0 0 0 60 0 0 0 0 0 0 17 18 0 0 0 0 45 44 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 50 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 49 31 0 0 0 0 19 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,49,19,0,0,0,0,31,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;`

C30.C23 in GAP, Magma, Sage, TeX

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

`G:=Group("C30.C2^3");`
`// GroupNames label`

`G:=SmallGroup(240,141);`
`// by ID`

`G=gap.SmallGroup(240,141);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^2=a^15,b*a*b=a^19,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d=a^15*b,d*c*d=a^15*c>;`
`// generators/relations`

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