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## G = C33⋊15D4order 216 = 23·33

### 3rd semidirect product of C33 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊15D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C33⋊15D4
 Lower central C33 — C32×C6 — C33⋊15D4
 Upper central C1 — C2 — C22

Generators and relations for C3315D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1060 in 224 conjugacy classes, 87 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, D4, C32, Dic3, D6, C2×C6, C3⋊S3, C3×C6, C3×C6, C3⋊D4, C33, C3⋊Dic3, C2×C3⋊S3, C62, C33⋊C2, C32×C6, C32×C6, C327D4, C335C4, C2×C33⋊C2, C3×C62, C3315D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, C3⋊D4, C2×C3⋊S3, C33⋊C2, C327D4, C2×C33⋊C2, C3315D4

Smallest permutation representation of C3315D4
On 108 points
Generators in S108
```(1 9 83)(2 84 10)(3 11 81)(4 82 12)(5 31 17)(6 18 32)(7 29 19)(8 20 30)(13 62 38)(14 39 63)(15 64 40)(16 37 61)(21 44 36)(22 33 41)(23 42 34)(24 35 43)(25 67 99)(26 100 68)(27 65 97)(28 98 66)(45 87 53)(46 54 88)(47 85 55)(48 56 86)(49 103 59)(50 60 104)(51 101 57)(52 58 102)(69 79 93)(70 94 80)(71 77 95)(72 96 78)(73 91 106)(74 107 92)(75 89 108)(76 105 90)
(1 70 42)(2 43 71)(3 72 44)(4 41 69)(5 40 67)(6 68 37)(7 38 65)(8 66 39)(9 94 34)(10 35 95)(11 96 36)(12 33 93)(13 97 29)(14 30 98)(15 99 31)(16 32 100)(17 64 25)(18 26 61)(19 62 27)(20 28 63)(21 81 78)(22 79 82)(23 83 80)(24 77 84)(45 106 58)(46 59 107)(47 108 60)(48 57 105)(49 92 54)(50 55 89)(51 90 56)(52 53 91)(73 102 87)(74 88 103)(75 104 85)(76 86 101)
(1 40 101)(2 102 37)(3 38 103)(4 104 39)(5 86 42)(6 43 87)(7 88 44)(8 41 85)(9 15 57)(10 58 16)(11 13 59)(12 60 14)(17 56 23)(18 24 53)(19 54 21)(20 22 55)(25 90 80)(26 77 91)(27 92 78)(28 79 89)(29 46 36)(30 33 47)(31 48 34)(32 35 45)(49 81 62)(50 63 82)(51 83 64)(52 61 84)(65 74 72)(66 69 75)(67 76 70)(68 71 73)(93 108 98)(94 99 105)(95 106 100)(96 97 107)
(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)
(2 4)(5 76)(6 75)(7 74)(8 73)(9 83)(10 82)(11 81)(12 84)(13 49)(14 52)(15 51)(16 50)(17 105)(18 108)(19 107)(20 106)(21 96)(22 95)(23 94)(24 93)(25 48)(26 47)(27 46)(28 45)(29 92)(30 91)(31 90)(32 89)(33 77)(34 80)(35 79)(36 78)(37 104)(38 103)(39 102)(40 101)(41 71)(42 70)(43 69)(44 72)(53 98)(54 97)(55 100)(56 99)(57 64)(58 63)(59 62)(60 61)(65 88)(66 87)(67 86)(68 85)```

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

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

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

C3315D4 is a maximal subgroup of   C62.90D6  C62.93D6  S3×C327D4  C3⋊S3×C3⋊D4  C62.160D6  D4×C33⋊C2  C62.100D6
C3315D4 is a maximal quotient of   C62.146D6  C62.148D6  C3315D8  C3324SD16  C3327SD16  C3315Q16  C63.C2

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 S3 D4 D6 C3⋊D4 kernel C33⋊15D4 C33⋊5C4 C2×C33⋊C2 C3×C62 C62 C33 C3×C6 C32 # reps 1 1 1 1 13 1 13 26

Matrix representation of C3315D4 in GL6(𝔽13)

 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1
,
 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
,
 11 9 0 0 0 0 11 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 4 11 0 0 0 0 2 9
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C3315D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_{15}D_4`
`% in TeX`

`G:=Group("C3^3:15D4");`
`// GroupNames label`

`G:=SmallGroup(216,149);`
`// by ID`

`G=gap.SmallGroup(216,149);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,1444,5189]);`
`// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

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