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## G = C33⋊12D4order 216 = 23·33

### 3rd semidirect product of C33 and D4 acting via D4/C4=C2

Aliases: C3312D4, C328D12, (C3×C12)⋊5S3, C121(C3⋊S3), C4⋊(C33⋊C2), (C3×C6).64D6, (C32×C12)⋊1C2, C31(C12⋊S3), (C32×C6).28C22, C6.16(C2×C3⋊S3), (C2×C33⋊C2)⋊3C2, C2.4(C2×C33⋊C2), SmallGroup(216,147)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊12D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C33⋊12D4
 Lower central C33 — C32×C6 — C33⋊12D4
 Upper central C1 — C2 — C4

Generators and relations for C3312D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1372 in 224 conjugacy classes, 87 normal (7 characteristic)
C1, C2, C2 [×2], C3 [×13], C4, C22 [×2], S3 [×26], C6 [×13], D4, C32 [×13], C12 [×13], D6 [×26], C3⋊S3 [×26], C3×C6 [×13], D12 [×13], C33, C3×C12 [×13], C2×C3⋊S3 [×26], C33⋊C2 [×2], C32×C6, C12⋊S3 [×13], C32×C12, C2×C33⋊C2 [×2], C3312D4
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], C3⋊S3 [×13], D12 [×13], C2×C3⋊S3 [×13], C33⋊C2, C12⋊S3 [×13], C2×C33⋊C2, C3312D4

Smallest permutation representation of C3312D4
On 108 points
Generators in S108
```(1 27 74)(2 28 75)(3 25 76)(4 26 73)(5 101 41)(6 102 42)(7 103 43)(8 104 44)(9 81 47)(10 82 48)(11 83 45)(12 84 46)(13 61 49)(14 62 50)(15 63 51)(16 64 52)(17 68 55)(18 65 56)(19 66 53)(20 67 54)(21 70 35)(22 71 36)(23 72 33)(24 69 34)(29 99 38)(30 100 39)(31 97 40)(32 98 37)(57 78 90)(58 79 91)(59 80 92)(60 77 89)(85 105 95)(86 106 96)(87 107 93)(88 108 94)
(1 49 31)(2 50 32)(3 51 29)(4 52 30)(5 80 56)(6 77 53)(7 78 54)(8 79 55)(9 93 22)(10 94 23)(11 95 24)(12 96 21)(13 97 27)(14 98 28)(15 99 25)(16 100 26)(17 104 91)(18 101 92)(19 102 89)(20 103 90)(33 48 108)(34 45 105)(35 46 106)(36 47 107)(37 75 62)(38 76 63)(39 73 64)(40 74 61)(41 59 65)(42 60 66)(43 57 67)(44 58 68)(69 83 85)(70 84 86)(71 81 87)(72 82 88)
(1 22 43)(2 23 44)(3 24 41)(4 21 42)(5 25 69)(6 26 70)(7 27 71)(8 28 72)(9 57 49)(10 58 50)(11 59 51)(12 60 52)(13 81 78)(14 82 79)(15 83 80)(16 84 77)(17 37 108)(18 38 105)(19 39 106)(20 40 107)(29 95 65)(30 96 66)(31 93 67)(32 94 68)(33 104 75)(34 101 76)(35 102 73)(36 103 74)(45 92 63)(46 89 64)(47 90 61)(48 91 62)(53 100 86)(54 97 87)(55 98 88)(56 99 85)
(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 4)(2 3)(5 33)(6 36)(7 35)(8 34)(9 66)(10 65)(11 68)(12 67)(13 39)(14 38)(15 37)(16 40)(17 83)(18 82)(19 81)(20 84)(21 43)(22 42)(23 41)(24 44)(25 75)(26 74)(27 73)(28 76)(29 50)(30 49)(31 52)(32 51)(45 55)(46 54)(47 53)(48 56)(57 96)(58 95)(59 94)(60 93)(61 100)(62 99)(63 98)(64 97)(69 104)(70 103)(71 102)(72 101)(77 107)(78 106)(79 105)(80 108)(85 91)(86 90)(87 89)(88 92)```

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

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

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

C3312D4 is a maximal subgroup of
C337D8  C338D8  C3315SD16  C3317SD16  C3321SD16  C3312D8  C3315D8  C3327SD16  C12.40S32  C12.58S32  S3×C12⋊S3  C3⋊S3×D12  C62.160D6  D4×C33⋊C2  (Q8×C33)⋊C2
C3312D4 is a maximal quotient of
C3321SD16  C3312D8  C3312Q16  C62.147D6  C62.148D6

57 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3M 4 6A ··· 6M 12A ··· 12Z order 1 2 2 2 3 ··· 3 4 6 ··· 6 12 ··· 12 size 1 1 54 54 2 ··· 2 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 S3 D4 D6 D12 kernel C33⋊12D4 C32×C12 C2×C33⋊C2 C3×C12 C33 C3×C6 C32 # reps 1 1 2 13 1 13 26

Matrix representation of C3312D4 in GL6(𝔽13)

 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 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 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 6 0 0 0 0 7 3
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 7 3 0 0 0 0 10 6

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

C3312D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_{12}D_4`
`% in TeX`

`G:=Group("C3^3:12D4");`
`// GroupNames label`

`G:=SmallGroup(216,147);`
`// by ID`

`G=gap.SmallGroup(216,147);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,5189]);`
`// Polycyclic`

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

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