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## G = C33⋊5C4order 108 = 22·33

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

Aliases: C335C4, C325Dic3, C3⋊(C3⋊Dic3), C6.3(C3⋊S3), (C3×C6).10S3, C2.(C33⋊C2), (C32×C6).3C2, SmallGroup(108,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊5C4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C33⋊5C4
 Lower central C33 — C33⋊5C4
 Upper central C1 — C2

Generators and relations for C335C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 240 in 84 conjugacy classes, 57 normal (5 characteristic)
C1, C2, C3, C4, C6, C32, Dic3, C3×C6, C33, C3⋊Dic3, C32×C6, C335C4
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, C3⋊Dic3, C33⋊C2, C335C4

Character table of C335C4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 27 27 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 2 -1 orthogonal lifted from S3 ρ6 2 2 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 2 -1 0 0 -1 2 -1 2 -1 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 0 0 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 2 orthogonal lifted from S3 ρ8 2 2 -1 -1 2 -1 -1 2 -1 2 -1 2 -1 -1 -1 0 0 -1 2 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 2 -1 2 -1 -1 -1 2 2 -1 -1 -1 2 -1 -1 0 0 2 -1 -1 -1 2 2 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -1 2 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 0 0 2 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ12 2 2 -1 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 0 0 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 0 0 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 orthogonal lifted from S3 ρ14 2 2 -1 2 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 0 0 2 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 0 0 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ16 2 2 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 0 0 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ17 2 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 0 0 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ18 2 -2 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 2 -1 0 0 1 -2 1 -2 1 -2 1 1 1 1 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ19 2 -2 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 0 0 1 1 -2 -2 -2 1 1 1 1 1 1 -2 1 symplectic lifted from Dic3, Schur index 2 ρ20 2 -2 -1 2 -1 -1 -1 2 2 -1 -1 -1 2 -1 -1 0 0 -2 1 1 1 -2 -2 1 1 1 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ21 2 -2 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 0 0 1 1 1 1 -2 1 1 -2 1 1 -2 1 -2 symplectic lifted from Dic3, Schur index 2 ρ22 2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 2 0 0 1 1 1 1 1 1 1 1 -2 -2 -2 -2 1 symplectic lifted from Dic3, Schur index 2 ρ23 2 -2 -1 2 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 0 0 -2 1 1 -2 1 1 1 -2 -2 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ24 2 -2 -1 -1 2 -1 -1 2 -1 2 -1 2 -1 -1 -1 0 0 1 -2 1 1 -2 1 -2 1 -2 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ25 2 -2 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 1 1 -2 1 1 -2 1 1 -2 1 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ26 2 -2 -1 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 0 0 1 -2 -2 1 1 1 1 -2 1 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ27 2 -2 2 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 0 0 1 1 1 -2 1 1 -2 1 1 -2 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ28 2 -2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 0 0 1 1 1 1 1 -2 -2 -2 1 1 1 -2 1 symplectic lifted from Dic3, Schur index 2 ρ29 2 -2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 0 0 -2 -2 1 1 1 1 1 1 1 1 1 -2 -2 symplectic lifted from Dic3, Schur index 2 ρ30 2 -2 -1 2 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 0 0 -2 1 -2 1 1 1 -2 1 1 1 -2 1 1 symplectic lifted from Dic3, Schur index 2

Smallest permutation representation of C335C4
Regular action on 108 points
Generators in S108
```(1 27 74)(2 75 28)(3 25 76)(4 73 26)(5 101 41)(6 42 102)(7 103 43)(8 44 104)(9 81 47)(10 48 82)(11 83 45)(12 46 84)(13 61 49)(14 50 62)(15 63 51)(16 52 64)(17 55 68)(18 65 56)(19 53 66)(20 67 54)(21 35 70)(22 71 36)(23 33 72)(24 69 34)(29 99 38)(30 39 100)(31 97 40)(32 37 98)(57 78 90)(58 91 79)(59 80 92)(60 89 77)(85 105 95)(86 96 106)(87 107 93)(88 94 108)
(1 49 31)(2 32 50)(3 51 29)(4 30 52)(5 80 56)(6 53 77)(7 78 54)(8 55 79)(9 93 22)(10 23 94)(11 95 24)(12 21 96)(13 97 27)(14 28 98)(15 99 25)(16 26 100)(17 91 104)(18 101 92)(19 89 102)(20 103 90)(33 108 48)(34 45 105)(35 106 46)(36 47 107)(37 62 75)(38 76 63)(39 64 73)(40 74 61)(41 59 65)(42 66 60)(43 57 67)(44 68 58)(69 83 85)(70 86 84)(71 81 87)(72 88 82)
(1 22 43)(2 44 23)(3 24 41)(4 42 21)(5 25 69)(6 70 26)(7 27 71)(8 72 28)(9 57 49)(10 50 58)(11 59 51)(12 52 60)(13 81 78)(14 79 82)(15 83 80)(16 77 84)(17 108 37)(18 38 105)(19 106 39)(20 40 107)(29 95 65)(30 66 96)(31 93 67)(32 68 94)(33 75 104)(34 101 76)(35 73 102)(36 103 74)(45 92 63)(46 64 89)(47 90 61)(48 62 91)(53 86 100)(54 97 87)(55 88 98)(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)```

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

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

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

C335C4 is a maximal subgroup of
S3×C3⋊Dic3  Dic3×C3⋊S3  C336D4  C334Q8  C338Q8  C4×C33⋊C2  C3315D4  C33⋊C12  C334C12  C325Dic9  C348C4  C324CSU2(𝔽3)  C6210Dic3
C335C4 is a maximal quotient of
C337C8  C325Dic9  He36Dic3  C348C4  C6210Dic3

Matrix representation of C335C4 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 9 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 5 0 0 0 0 0 0 0 3 6 0 0 0 0 0 3 10 0 0 0 0 0 0 0 5 5 0 0 0 0 0 3 8 0 0 0 0 0 0 0 10 7 0 0 0 0 0 10 3

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

C335C4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_5C_4`
`% in TeX`

`G:=Group("C3^3:5C4");`
`// GroupNames label`

`G:=SmallGroup(108,34);`
`// by ID`

`G=gap.SmallGroup(108,34);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-3,10,122,483,1804]);`
`// Polycyclic`

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

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