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## G = C32⋊F7order 378 = 2·33·7

### The semidirect product of C32 and F7 acting via F7/C7=C6

Aliases: C32⋊F7, C7⋊He31C2, C7⋊(C32⋊C6), (C3×C21)⋊3C6, C3⋊D212C3, C21.4(C3×S3), C3.4(C3⋊F7), (C3×C7⋊C3)⋊1S3, SmallGroup(378,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — C32⋊F7
 Chief series C1 — C7 — C21 — C3×C21 — C7⋊He3 — C32⋊F7
 Lower central C3×C21 — C32⋊F7
 Upper central C1

Generators and relations for C32⋊F7
G = < a,b,c,d | a3=b3=c7=d6=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=b-1, dcd-1=c5 >

63C2
3C3
21C3
42C3
21S3
63C6
63S3
7C32
14C32
9D7
3C21
21C3×S3
7He3
3D21
9F7
9D21

Character table of C32⋊F7

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 7 21A 21B 21C 21D 21E 21F 21G 21H size 1 63 2 6 21 21 42 42 63 63 6 6 6 6 6 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 2 0 2 -1 2 2 -1 -1 0 0 2 -1 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 -1 2 2 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 0 2 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 -1 2 2 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 6 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ11 6 0 -3 0 0 0 0 0 0 0 6 0 -3 -3 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ12 6 0 6 -3 0 0 0 0 0 0 -1 1-√21/2 -1 -1 1-√21/2 1+√21/2 1+√21/2 1-√21/2 1+√21/2 orthogonal lifted from C3⋊F7 ρ13 6 0 -3 0 0 0 0 0 0 0 -1 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 1+√21/2 1-√21/2 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 orthogonal faithful ρ14 6 0 -3 0 0 0 0 0 0 0 -1 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 1-√21/2 1+√21/2 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 orthogonal faithful ρ15 6 0 -3 0 0 0 0 0 0 0 -1 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 1-√21/2 1+√21/2 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 orthogonal faithful ρ16 6 0 -3 0 0 0 0 0 0 0 -1 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 1+√21/2 1-√21/2 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 orthogonal faithful ρ17 6 0 -3 0 0 0 0 0 0 0 -1 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 1-√21/2 1+√21/2 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 orthogonal faithful ρ18 6 0 6 -3 0 0 0 0 0 0 -1 1+√21/2 -1 -1 1+√21/2 1-√21/2 1-√21/2 1+√21/2 1-√21/2 orthogonal lifted from C3⋊F7 ρ19 6 0 -3 0 0 0 0 0 0 0 -1 2ζ32ζ76+ζ32ζ74+ζ32ζ73+2ζ32ζ72+ζ32+ζ76+2ζ74+2ζ73+ζ72+1 1+√21/2 1-√21/2 ζ32ζ75+2ζ32ζ73+ζ32ζ72+2ζ32ζ7+ζ32+2ζ75+ζ73+2ζ72+ζ7+1 ζ3ζ75+2ζ3ζ73+ζ3ζ72+2ζ3ζ7+ζ3+2ζ75+ζ73+2ζ72+ζ7+1 2ζ32ζ75+ζ32ζ74+ζ32ζ73+2ζ32ζ7+ζ32+ζ75+2ζ74+2ζ73+ζ7+1 ζ3ζ76+2ζ3ζ73+2ζ3ζ72+ζ3ζ7+ζ3+2ζ76+ζ73+ζ72+2ζ7+1 ζ3ζ76+2ζ3ζ75+2ζ3ζ74+ζ3ζ7+ζ3+2ζ76+ζ75+ζ74+2ζ7+1 orthogonal faithful

Smallest permutation representation of C32⋊F7
On 63 points
Generators in S63
```(1 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(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)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)(22 50 29 43 36 57)(23 53 31 49 40 62)(24 56 33 48 37 60)(25 52 35 47 41 58)(26 55 30 46 38 63)(27 51 32 45 42 61)(28 54 34 44 39 59)```

`G:=sub<Sym(63)| (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,50,29,43,36,57)(23,53,31,49,40,62)(24,56,33,48,37,60)(25,52,35,47,41,58)(26,55,30,46,38,63)(27,51,32,45,42,61)(28,54,34,44,39,59)>;`

`G:=Group( (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,50,29,43,36,57)(23,53,31,49,40,62)(24,56,33,48,37,60)(25,52,35,47,41,58)(26,55,30,46,38,63)(27,51,32,45,42,61)(28,54,34,44,39,59) );`

`G=PermutationGroup([[(1,43,22),(2,44,23),(3,45,24),(4,46,25),(5,47,26),(6,48,27),(7,49,28),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(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)], [(2,4,3,7,5,6),(8,15),(9,18,10,21,12,20),(11,17,14,19,13,16),(22,50,29,43,36,57),(23,53,31,49,40,62),(24,56,33,48,37,60),(25,52,35,47,41,58),(26,55,30,46,38,63),(27,51,32,45,42,61),(28,54,34,44,39,59)]])`

Matrix representation of C32⋊F7 in GL6(𝔽43)

 40 33 15 30 29 4 39 36 29 11 26 25 18 14 11 4 29 1 42 17 13 10 3 28 15 14 32 28 25 18 25 40 39 14 10 7
,
 2 5 5 0 5 0 0 2 5 5 0 5 38 38 40 0 0 38 5 0 0 2 5 5 38 0 38 38 40 0 0 38 0 38 38 40
,
 42 42 42 42 42 42 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 1 0
,
 42 0 0 0 0 0 0 0 0 0 0 42 0 0 0 42 0 0 0 42 0 0 0 0 1 1 1 1 1 1 0 0 0 0 42 0

`G:=sub<GL(6,GF(43))| [40,39,18,42,15,25,33,36,14,17,14,40,15,29,11,13,32,39,30,11,4,10,28,14,29,26,29,3,25,10,4,25,1,28,18,7],[2,0,38,5,38,0,5,2,38,0,0,38,5,5,40,0,38,0,0,5,0,2,38,38,5,0,0,5,40,38,0,5,38,5,0,40],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[42,0,0,0,1,0,0,0,0,42,1,0,0,0,0,0,1,0,0,0,42,0,1,0,0,0,0,0,1,42,0,42,0,0,1,0] >;`

C32⋊F7 in GAP, Magma, Sage, TeX

`C_3^2\rtimes F_7`
`% in TeX`

`G:=Group("C3^2:F7");`
`// GroupNames label`

`G:=SmallGroup(378,22);`
`// by ID`

`G=gap.SmallGroup(378,22);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-7,182,187,723,8104,1359]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^7=d^6=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;`
`// generators/relations`

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