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## G = C33⋊4C12order 324 = 22·34

### 4th semidirect product of C33 and C12 acting via C12/C2=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊4C12
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×He3 — C33⋊4C12
 Lower central C33 — C33⋊4C12
 Upper central C1 — C2

Generators and relations for C334C12
G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, dad-1=a-1c, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 468 in 108 conjugacy classes, 34 normal (16 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C33, C3×Dic3, C3⋊Dic3, C2×He3, C2×He3, C32×C6, C32×C6, C3×He3, C32⋊C12, C3×C3⋊Dic3, C335C4, C6×He3, C334C12
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C32⋊C6, C3×C3⋊S3, C32⋊C12, C3×C3⋊Dic3, He34S3, C334C12

Smallest permutation representation of C334C12
On 108 points
Generators in S108
```(1 84 95)(2 15 104)(3 47 35)(4 86 75)(5 107 18)(6 26 38)(7 78 89)(8 21 98)(9 41 29)(10 92 81)(11 101 24)(12 32 44)(13 83 59)(14 62 45)(16 50 74)(17 48 65)(19 77 53)(20 68 39)(22 56 80)(23 42 71)(25 66 76)(27 54 97)(28 79 69)(30 100 57)(31 72 82)(33 60 103)(34 73 63)(36 106 51)(37 87 52)(40 55 90)(43 93 58)(46 49 96)(61 94 102)(64 105 85)(67 88 108)(70 99 91)
(1 91 107)(2 108 92)(3 93 97)(4 98 94)(5 95 99)(6 100 96)(7 85 101)(8 102 86)(9 87 103)(10 104 88)(11 89 105)(12 106 90)(13 65 79)(14 80 66)(15 67 81)(16 82 68)(17 69 83)(18 84 70)(19 71 73)(20 74 72)(21 61 75)(22 76 62)(23 63 77)(24 78 64)(25 45 56)(26 57 46)(27 47 58)(28 59 48)(29 37 60)(30 49 38)(31 39 50)(32 51 40)(33 41 52)(34 53 42)(35 43 54)(36 55 44)
(1 62 60)(2 49 63)(3 64 50)(4 51 65)(5 66 52)(6 53 67)(7 68 54)(8 55 69)(9 70 56)(10 57 71)(11 72 58)(12 59 61)(13 94 32)(14 33 95)(15 96 34)(16 35 85)(17 86 36)(18 25 87)(19 88 26)(20 27 89)(21 90 28)(22 29 91)(23 92 30)(24 31 93)(37 107 76)(38 77 108)(39 97 78)(40 79 98)(41 99 80)(42 81 100)(43 101 82)(44 83 102)(45 103 84)(46 73 104)(47 105 74)(48 75 106)
(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)```

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G ··· 3Q 4A 4B 6A 6B 6C 6D 6E 6F 6G ··· 6Q 12A 12B 12C 12D order 1 2 3 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 6 6 ··· 6 12 12 12 12 size 1 1 2 2 2 2 3 3 6 ··· 6 27 27 2 2 2 2 3 3 6 ··· 6 27 27 27 27

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + - - + - image C1 C2 C3 C4 C6 C12 S3 S3 Dic3 Dic3 C3×S3 C3×Dic3 C32⋊C6 C32⋊C12 kernel C33⋊4C12 C6×He3 C33⋊5C4 C3×He3 C32×C6 C33 C2×He3 C32×C6 He3 C33 C3×C6 C32 C6 C3 # reps 1 1 2 2 2 4 3 1 3 1 8 8 3 3

Matrix representation of C334C12 in GL8(𝔽13)

 9 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 4 4 10 10 11 12 0 0 0 0 0 0 3 0 0 0 1 0 0 0 3 0
,
 3 0 0 0 0 0 0 0 3 9 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 9 0 3 0 0 1 0 0 0 4 0 10 12 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 9 0 3 0 0 1 0 0 0 4 0 10 12 12
,
 2 4 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 3 7 6 4 4 2 0 0 0 2 11 0 2 11 0 0 7 5 8 6 4 2 0 0 2 11 2 11 2 11 0 0 12 11 11 6 4 2 0 0 5 8 7 0 2 11

`G:=sub<GL(8,GF(13))| [9,10,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,4,0,0,0,0,1,0,0,10,0,0,0,0,0,1,0,10,0,0,0,0,0,0,12,11,3,3,0,0,0,0,1,12,0,0],[3,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,12,12,0,0,9,0,0,0,1,0,0,0,0,4,0,0,0,0,12,12,3,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,9,0,0,0,1,0,0,0,0,4,0,0,0,0,12,12,3,0,0,0,0,0,1,0,0,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[2,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,3,0,7,2,12,5,0,0,7,2,5,11,11,8,0,0,6,11,8,2,11,7,0,0,4,0,6,11,6,0,0,0,4,2,4,2,4,2,0,0,2,11,2,11,2,11] >;`

C334C12 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_4C_{12}`
`% in TeX`

`G:=Group("C3^3:4C12");`
`// GroupNames label`

`G:=SmallGroup(324,98);`
`// by ID`

`G=gap.SmallGroup(324,98);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,1449,2164,7781]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*c,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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