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## G = C42⋊3- 1+2order 432 = 24·33

### The semidirect product of C42 and 3- 1+2 acting via 3- 1+2/C9=C3

Aliases: C4213- 1+2, C9⋊(C42⋊C3), (C4×C36)⋊2C3, C42⋊C91C3, (C2×C18).3A4, C22.(C9⋊A4), (C4×C12).2C32, (C3×C42⋊C3).C3, (C2×C6).7(C3×A4), C3.3(C3×C42⋊C3), SmallGroup(432,100)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C12 — C42⋊3- 1+2
 Chief series C1 — C22 — C42 — C4×C12 — C3×C42⋊C3 — C42⋊3- 1+2
 Lower central C42 — C4×C12 — C42⋊3- 1+2
 Upper central C1 — C3 — C9

Generators and relations for C42⋊3- 1+2
G = < a,b,c,d | a4=b4=c9=d3=1, dbd-1=ab=ba, cac-1=ab-1, dad-1=a2b, cbc-1=a-1b2, dcd-1=c4 >

Smallest permutation representation of C42⋊3- 1+2
On 108 points
Generators in S108
```(1 40)(2 104 41 63)(3 105 42 55)(4 43)(5 107 44 57)(6 108 45 58)(7 37)(8 101 38 60)(9 102 39 61)(10 82)(11 66 83 94)(12 67 84 95)(13 85)(14 69 86 97)(15 70 87 98)(16 88)(17 72 89 91)(18 64 90 92)(19 54 77 28)(20 78)(21 47 79 30)(22 48 80 31)(23 81)(24 50 73 33)(25 51 74 34)(26 75)(27 53 76 36)(29 46)(32 49)(35 52)(56 106)(59 100)(62 103)(65 93)(68 96)(71 99)
(1 103 40 62)(3 55 42 105)(4 106 43 56)(6 58 45 108)(7 100 37 59)(9 61 39 102)(10 65 82 93)(12 95 84 67)(13 68 85 96)(15 98 87 70)(16 71 88 99)(18 92 90 64)(19 28 77 54)(20 46 78 29)(22 31 80 48)(23 49 81 32)(25 34 74 51)(26 52 75 35)
(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)
(1 31 94)(2 29 98)(3 36 93)(4 34 97)(5 32 92)(6 30 96)(7 28 91)(8 35 95)(9 33 99)(10 105 27)(11 103 22)(12 101 26)(13 108 21)(14 106 25)(15 104 20)(16 102 24)(17 100 19)(18 107 23)(37 54 72)(38 52 67)(39 50 71)(40 48 66)(41 46 70)(42 53 65)(43 51 69)(44 49 64)(45 47 68)(55 76 82)(56 74 86)(57 81 90)(58 79 85)(59 77 89)(60 75 84)(61 73 88)(62 80 83)(63 78 87)```

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

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

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

56 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 9A 9B 9C 9D 9E 9F 12A ··· 12H 18A ··· 18F 36A ··· 36X order 1 2 3 3 3 3 4 4 4 4 6 6 9 9 9 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 3 1 1 48 48 3 3 3 3 3 3 3 3 48 48 48 48 3 ··· 3 3 ··· 3 3 ··· 3

56 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 3 type + + image C1 C3 C3 C3 A4 3- 1+2 C3×A4 C42⋊C3 C9⋊A4 C3×C42⋊C3 C42⋊3- 1+2 kernel C42⋊3- 1+2 C42⋊C9 C4×C36 C3×C42⋊C3 C2×C18 C42 C2×C6 C9 C22 C3 C1 # reps 1 4 2 2 1 2 2 4 6 8 24

Matrix representation of C42⋊3- 1+2 in GL3(𝔽37) generated by

 36 0 0 0 6 0 0 0 6
,
 6 0 0 0 31 0 0 0 1
,
 0 7 0 0 0 33 34 0 0
,
 0 0 1 1 0 0 0 1 0
`G:=sub<GL(3,GF(37))| [36,0,0,0,6,0,0,0,6],[6,0,0,0,31,0,0,0,1],[0,0,34,7,0,0,0,33,0],[0,1,0,0,0,1,1,0,0] >;`

C42⋊3- 1+2 in GAP, Magma, Sage, TeX

`C_4^2\rtimes 3_-^{1+2}`
`% in TeX`

`G:=Group("C4^2:ES-(3,1)");`
`// GroupNames label`

`G:=SmallGroup(432,100);`
`// by ID`

`G=gap.SmallGroup(432,100);`
`# by ID`

`G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,50,1515,360,10399,102,9077,15882]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^9=d^3=1,d*b*d^-1=a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^2*b,c*b*c^-1=a^-1*b^2,d*c*d^-1=c^4>;`
`// generators/relations`

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