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## G = C3⋊S3×C7⋊C3order 378 = 2·33·7

### Direct product of C3⋊S3 and C7⋊C3

Aliases: C3⋊S3×C7⋊C3, C213(C3×S3), (C3×C21)⋊11C6, C3⋊(S3×C7⋊C3), C72(C3×C3⋊S3), (C3×C7⋊C3)⋊5S3, (C7×C3⋊S3)⋊3C3, (C32×C7⋊C3)⋊5C2, C324(C2×C7⋊C3), SmallGroup(378,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — C3⋊S3×C7⋊C3
 Chief series C1 — C7 — C21 — C3×C21 — C32×C7⋊C3 — C3⋊S3×C7⋊C3
 Lower central C3×C21 — C3⋊S3×C7⋊C3
 Upper central C1

Generators and relations for C3⋊S3×C7⋊C3
G = < a,b,c,d,e | a3=b3=c2=d7=e3=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 408 in 64 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3, C3, S3, C6, C7, C32, C32, C14, C3×S3, C3⋊S3, C7⋊C3, C7⋊C3, C21, C33, C2×C7⋊C3, S3×C7, C3×C3⋊S3, C3×C7⋊C3, C3×C7⋊C3, C3×C21, S3×C7⋊C3, C7×C3⋊S3, C32×C7⋊C3, C3⋊S3×C7⋊C3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C7⋊C3, C2×C7⋊C3, C3×C3⋊S3, S3×C7⋊C3, C3⋊S3×C7⋊C3

Character table of C3⋊S3×C7⋊C3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 6A 6B 7A 7B 14A 14B 21A 21B 21C 21D 21E 21F 21G 21H size 1 9 2 2 2 2 7 7 14 14 14 14 14 14 14 14 63 63 3 3 27 27 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 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 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ6 ζ65 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ65 ζ6 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ7 2 0 -1 -1 2 -1 2 2 2 -1 -1 -1 2 -1 -1 -1 0 0 2 2 0 0 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 2 -1 -1 2 2 -1 -1 -1 2 -1 -1 -1 2 0 0 2 2 0 0 2 -1 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ9 2 0 -1 -1 -1 2 2 2 -1 2 -1 -1 -1 2 -1 -1 0 0 2 2 0 0 -1 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 -1 -1 -1 2 2 -1 -1 2 -1 -1 -1 2 -1 0 0 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ11 2 0 -1 2 -1 -1 -1-√-3 -1+√-3 ζ65 ζ6 ζ6 -1-√-3 ζ6 ζ65 ζ65 -1+√-3 0 0 2 2 0 0 2 -1 -1 -1 -1 -1 2 -1 complex lifted from C3×S3 ρ12 2 0 2 -1 -1 -1 -1-√-3 -1+√-3 ζ65 ζ6 -1-√-3 ζ6 ζ6 ζ65 -1+√-3 ζ65 0 0 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 complex lifted from C3×S3 ρ13 2 0 -1 2 -1 -1 -1+√-3 -1-√-3 ζ6 ζ65 ζ65 -1+√-3 ζ65 ζ6 ζ6 -1-√-3 0 0 2 2 0 0 2 -1 -1 -1 -1 -1 2 -1 complex lifted from C3×S3 ρ14 2 0 -1 -1 2 -1 -1+√-3 -1-√-3 -1-√-3 ζ65 ζ65 ζ65 -1+√-3 ζ6 ζ6 ζ6 0 0 2 2 0 0 -1 -1 -1 -1 2 2 -1 -1 complex lifted from C3×S3 ρ15 2 0 2 -1 -1 -1 -1+√-3 -1-√-3 ζ6 ζ65 -1+√-3 ζ65 ζ65 ζ6 -1-√-3 ζ6 0 0 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 complex lifted from C3×S3 ρ16 2 0 -1 -1 -1 2 -1+√-3 -1-√-3 ζ6 -1+√-3 ζ65 ζ65 ζ65 -1-√-3 ζ6 ζ6 0 0 2 2 0 0 -1 2 2 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ17 2 0 -1 -1 2 -1 -1-√-3 -1+√-3 -1+√-3 ζ6 ζ6 ζ6 -1-√-3 ζ65 ζ65 ζ65 0 0 2 2 0 0 -1 -1 -1 -1 2 2 -1 -1 complex lifted from C3×S3 ρ18 2 0 -1 -1 -1 2 -1-√-3 -1+√-3 ζ65 -1-√-3 ζ6 ζ6 ζ6 -1+√-3 ζ65 ζ65 0 0 2 2 0 0 -1 2 2 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ19 3 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ20 3 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ21 3 -3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ22 3 -3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ23 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 -1-√-7 1+√-7/2 1-√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 -1+√-7 1+√-7/2 complex lifted from S3×C7⋊C3 ρ24 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1+√-7/2 -1-√-7 -1+√-7 1-√-7/2 1-√-7/2 1+√-7/2 1-√-7/2 1+√-7/2 complex lifted from S3×C7⋊C3 ρ25 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1+√-7/2 1+√-7/2 1-√-7/2 -1+√-7 1-√-7/2 1+√-7/2 1-√-7/2 -1-√-7 complex lifted from S3×C7⋊C3 ρ26 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 -1+√-7 1-√-7/2 1+√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 -1-√-7 1-√-7/2 complex lifted from S3×C7⋊C3 ρ27 6 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1-√-7/2 -1+√-7 -1-√-7 1+√-7/2 1+√-7/2 1-√-7/2 1+√-7/2 1-√-7/2 complex lifted from S3×C7⋊C3 ρ28 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1-√-7/2 1-√-7/2 1+√-7/2 -1-√-7 1+√-7/2 1-√-7/2 1+√-7/2 -1+√-7 complex lifted from S3×C7⋊C3 ρ29 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 -1-√-7 -1+√-7 1+√-7/2 1-√-7/2 complex lifted from S3×C7⋊C3 ρ30 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 -1+√-7 -1-√-7 1-√-7/2 1+√-7/2 complex lifted from S3×C7⋊C3

Smallest permutation representation of C3⋊S3×C7⋊C3
On 63 points
Generators in S63
(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 22 43)(2 23 44)(3 24 45)(4 25 46)(5 26 47)(6 27 48)(7 28 49)(8 29 50)(9 30 51)(10 31 52)(11 32 53)(12 33 54)(13 34 55)(14 35 56)(15 36 57)(16 37 58)(17 38 59)(18 39 60)(19 40 61)(20 41 62)(21 42 63)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 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 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)

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

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

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

Matrix representation of C3⋊S3×C7⋊C3 in GL7(𝔽43)

 0 1 0 0 0 0 0 42 42 0 0 0 0 0 0 0 41 37 0 0 0 0 0 22 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
,
 42 42 0 0 0 0 0 1 0 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 1
,
 1 0 0 0 0 0 0 42 42 0 0 0 0 0 0 0 1 0 0 0 0 0 0 21 42 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 24 25 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 42 42 0 0 0 0 0 1 0

G:=sub<GL(7,GF(43))| [0,42,0,0,0,0,0,1,42,0,0,0,0,0,0,0,41,22,0,0,0,0,0,37,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[42,1,0,0,0,0,0,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,42,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,21,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,24,1,0,0,0,0,0,25,0,1,0,0,0,0,1,0,0],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,42,1,0,0,0,0,0,42,0] >;

C3⋊S3×C7⋊C3 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_7\rtimes C_3
% in TeX

G:=Group("C3:S3xC7:C3");
// GroupNames label

G:=SmallGroup(378,50);
// by ID

G=gap.SmallGroup(378,50);
# by ID

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

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^7=e^3=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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