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## G = C32⋊3D20order 360 = 23·32·5

### 2nd semidirect product of C32 and D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C32⋊3D20
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — C32⋊3D20
 Lower central C3×C15 — C3×C30 — C32⋊3D20
 Upper central C1 — C2

Generators and relations for C323D20
G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 452 in 70 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, Dic3, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, D10, C3⋊D4, C3×D5, D15, C30, C30, C3⋊Dic3, S3×C6, D20, C3×C15, C5×Dic3, C6×D5, D30, D6⋊S3, C3×D15, C3×C30, C3⋊D20, C5×C3⋊Dic3, C6×D15, C323D20
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3⋊D4, S32, D20, S3×D5, D6⋊S3, C3⋊D20, D15⋊S3, C323D20

Smallest permutation representation of C323D20
On 120 points
Generators in S120
```(1 90 43)(2 44 91)(3 92 45)(4 46 93)(5 94 47)(6 48 95)(7 96 49)(8 50 97)(9 98 51)(10 52 99)(11 100 53)(12 54 81)(13 82 55)(14 56 83)(15 84 57)(16 58 85)(17 86 59)(18 60 87)(19 88 41)(20 42 89)(21 101 63)(22 64 102)(23 103 65)(24 66 104)(25 105 67)(26 68 106)(27 107 69)(28 70 108)(29 109 71)(30 72 110)(31 111 73)(32 74 112)(33 113 75)(34 76 114)(35 115 77)(36 78 116)(37 117 79)(38 80 118)(39 119 61)(40 62 120)
(1 43 90)(2 91 44)(3 45 92)(4 93 46)(5 47 94)(6 95 48)(7 49 96)(8 97 50)(9 51 98)(10 99 52)(11 53 100)(12 81 54)(13 55 82)(14 83 56)(15 57 84)(16 85 58)(17 59 86)(18 87 60)(19 41 88)(20 89 42)(21 101 63)(22 64 102)(23 103 65)(24 66 104)(25 105 67)(26 68 106)(27 107 69)(28 70 108)(29 109 71)(30 72 110)(31 111 73)(32 74 112)(33 113 75)(34 76 114)(35 115 77)(36 78 116)(37 117 79)(38 80 118)(39 119 61)(40 62 120)
(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 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 102)(42 101)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 2 2 3 3 3 4 5 5 6 6 6 6 6 6 6 10 10 15 ··· 15 20 20 20 20 30 ··· 30 size 1 1 30 30 2 2 4 18 2 2 2 2 4 30 30 30 30 2 2 4 ··· 4 18 18 18 18 4 ··· 4

39 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + - + image C1 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 D20 S32 S3×D5 D6⋊S3 C3⋊D20 D15⋊S3 C32⋊3D20 kernel C32⋊3D20 C5×C3⋊Dic3 C6×D15 D30 C3×C15 C3⋊Dic3 C30 C3×C6 C15 C32 C10 C6 C5 C3 C2 C1 # reps 1 1 2 2 1 2 2 2 4 4 1 4 1 4 4 4

Matrix representation of C323D20 in GL6(𝔽61)

 1 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 60 60 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 27 36 0 0 0 0 25 4 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 60 60
,
 52 45 0 0 0 0 5 9 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,25,0,0,0,0,36,4,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[52,5,0,0,0,0,45,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C323D20 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_3D_{20}`
`% in TeX`

`G:=Group("C3^2:3D20");`
`// GroupNames label`

`G:=SmallGroup(360,87);`
`// by ID`

`G=gap.SmallGroup(360,87);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,387,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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