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G = C3⋊S3×C7⋊C3  order 378 = 2·33·7

Direct product of C3⋊S3 and C7⋊C3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3⋊S3×C7⋊C3, C21⋊3(C3×S3), (C3×C21)⋊11C6, C3⋊(S3×C7⋊C3), C7⋊2(C3×C3⋊S3), (C3×C7⋊C3)⋊5S3, (C7×C3⋊S3)⋊3C3, (C32×C7⋊C3)⋊5C2, C32⋊4(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 123A3B3C3D3E3F3G3H3I3J3K3L3M3N6A6B7A7B14A14B21A21B21C21D21E21F21G21H
 size 192222771414141414141414636333272766666666
ρ1111111111111111111111111111111    trivial
ρ21-111111111111111-1-111-1-111111111    linear of order 2
ρ31-11111ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ6ζ6511-1-111111111    linear of order 6
ρ4111111ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3111111111111    linear of order 3
ρ5111111ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32111111111111    linear of order 3
ρ61-11111ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ65ζ611-1-111111111    linear of order 6
ρ720-1-12-1222-1-1-12-1-1-1002200-1-1-1-122-1-1    orthogonal lifted from S3
ρ820-12-1-122-1-1-12-1-1-120022002-1-1-1-1-12-1    orthogonal lifted from S3
ρ920-1-1-1222-12-1-1-12-1-1002200-122-1-1-1-1-1    orthogonal lifted from S3
ρ10202-1-1-122-1-12-1-1-12-1002200-1-1-12-1-1-12    orthogonal lifted from S3
ρ1120-12-1-1-1-√-3-1+√-3ζ65ζ6ζ6-1-√-3ζ6ζ65ζ65-1+√-30022002-1-1-1-1-12-1    complex lifted from C3×S3
ρ12202-1-1-1-1-√-3-1+√-3ζ65ζ6-1-√-3ζ6ζ6ζ65-1+√-3ζ65002200-1-1-12-1-1-12    complex lifted from C3×S3
ρ1320-12-1-1-1+√-3-1-√-3ζ6ζ65ζ65-1+√-3ζ65ζ6ζ6-1-√-30022002-1-1-1-1-12-1    complex lifted from C3×S3
ρ1420-1-12-1-1+√-3-1-√-3-1-√-3ζ65ζ65ζ65-1+√-3ζ6ζ6ζ6002200-1-1-1-122-1-1    complex lifted from C3×S3
ρ15202-1-1-1-1+√-3-1-√-3ζ6ζ65-1+√-3ζ65ζ65ζ6-1-√-3ζ6002200-1-1-12-1-1-12    complex lifted from C3×S3
ρ1620-1-1-12-1+√-3-1-√-3ζ6-1+√-3ζ65ζ65ζ65-1-√-3ζ6ζ6002200-122-1-1-1-1-1    complex lifted from C3×S3
ρ1720-1-12-1-1-√-3-1+√-3-1+√-3ζ6ζ6ζ6-1-√-3ζ65ζ65ζ65002200-1-1-1-122-1-1    complex lifted from C3×S3
ρ1820-1-1-12-1-√-3-1+√-3ζ65-1-√-3ζ6ζ6ζ6-1+√-3ζ65ζ65002200-122-1-1-1-1-1    complex lifted from C3×S3
ρ19333333000000000000-1-√-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
ρ20333333000000000000-1+√-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
ρ213-33333000000000000-1+√-7/2-1-√-7/21-√-7/21+√-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
ρ223-33333000000000000-1-√-7/2-1+√-7/21+√-7/21-√-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
ρ2360-36-3-3000000000000-1-√-7-1+√-700-1-√-71+√-7/21-√-7/21-√-7/21-√-7/21+√-7/2-1+√-71+√-7/2    complex lifted from S3×C7⋊C3
ρ2460-3-3-36000000000000-1-√-7-1+√-7001+√-7/2-1-√-7-1+√-71-√-7/21-√-7/21+√-7/21-√-7/21+√-7/2    complex lifted from S3×C7⋊C3
ρ25606-3-3-3000000000000-1-√-7-1+√-7001+√-7/21+√-7/21-√-7/2-1+√-71-√-7/21+√-7/21-√-7/2-1-√-7    complex lifted from S3×C7⋊C3
ρ2660-36-3-3000000000000-1+√-7-1-√-700-1+√-71-√-7/21+√-7/21+√-7/21+√-7/21-√-7/2-1-√-71-√-7/2    complex lifted from S3×C7⋊C3
ρ2760-3-3-36000000000000-1+√-7-1-√-7001-√-7/2-1+√-7-1-√-71+√-7/21+√-7/21-√-7/21+√-7/21-√-7/2    complex lifted from S3×C7⋊C3
ρ28606-3-3-3000000000000-1+√-7-1-√-7001-√-7/21-√-7/21+√-7/2-1-√-71+√-7/21-√-7/21+√-7/2-1+√-7    complex lifted from S3×C7⋊C3
ρ2960-3-36-3000000000000-1+√-7-1-√-7001-√-7/21-√-7/21+√-7/21+√-7/2-1-√-7-1+√-71+√-7/21-√-7/2    complex lifted from S3×C7⋊C3
ρ3060-3-36-3000000000000-1-√-7-1+√-7001+√-7/21+√-7/21-√-7/21-√-7/2-1+√-7-1-√-71-√-7/21+√-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)

0100000
424200000
004137000
00221000
0000100
0000010
0000001
,
424200000
1000000
0010000
0001000
0000100
0000010
0000001
,
1000000
424200000
0010000
002142000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
000024251
0000100
0000010
,
36000000
03600000
0060000
0006000
0000100
0000184242
0000010

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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Character table of C3⋊S3×C7⋊C3 in TeX

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