Copied to
clipboard

G = C6×C33⋊C2order 324 = 22·34

Direct product of C6 and C33⋊C2

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

Aliases: C6×C33⋊C2, C3320D6, C349C22, (C32×C6)⋊8S3, (C32×C6)⋊9C6, (C33×C6)⋊3C2, C3313(C2×C6), C3211(S3×C6), C6⋊(C3×C3⋊S3), C32(C6×C3⋊S3), (C3×C6)⋊6(C3×S3), (C3×C6)⋊3(C3⋊S3), C328(C2×C3⋊S3), SmallGroup(324,174)

Series: Derived Chief Lower central Upper central

C1C33 — C6×C33⋊C2
C1C3C32C33C34C3×C33⋊C2 — C6×C33⋊C2
C33 — C6×C33⋊C2
C1C6

Generators and relations for C6×C33⋊C2
 G = < a,b,c,d,e | a6=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1528 in 436 conjugacy classes, 118 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×13], C3 [×13], C22, S3 [×26], C6, C6 [×13], C6 [×15], C32 [×26], C32 [×52], D6 [×13], C2×C6, C3×S3 [×26], C3⋊S3 [×26], C3×C6 [×26], C3×C6 [×52], C33, C33 [×13], C33 [×13], S3×C6 [×13], C2×C3⋊S3 [×13], C3×C3⋊S3 [×26], C33⋊C2 [×2], C32×C6, C32×C6 [×13], C32×C6 [×13], C34, C6×C3⋊S3 [×13], C2×C33⋊C2, C3×C33⋊C2 [×2], C33×C6, C6×C33⋊C2
Quotients: C1, C2 [×3], C3, C22, S3 [×13], C6 [×3], D6 [×13], C2×C6, C3×S3 [×13], C3⋊S3 [×13], S3×C6 [×13], C2×C3⋊S3 [×13], C3×C3⋊S3 [×13], C33⋊C2, C6×C3⋊S3 [×13], C2×C33⋊C2, C3×C33⋊C2, C6×C33⋊C2

Smallest permutation representation of C6×C33⋊C2
On 108 points
Generators in S108
(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)
(1 101 41)(2 102 42)(3 97 37)(4 98 38)(5 99 39)(6 100 40)(7 68 92)(8 69 93)(9 70 94)(10 71 95)(11 72 96)(12 67 91)(13 46 90)(14 47 85)(15 48 86)(16 43 87)(17 44 88)(18 45 89)(19 63 52)(20 64 53)(21 65 54)(22 66 49)(23 61 50)(24 62 51)(25 108 81)(26 103 82)(27 104 83)(28 105 84)(29 106 79)(30 107 80)(31 74 57)(32 75 58)(33 76 59)(34 77 60)(35 78 55)(36 73 56)
(1 57 91)(2 58 92)(3 59 93)(4 60 94)(5 55 95)(6 56 96)(7 102 32)(8 97 33)(9 98 34)(10 99 35)(11 100 36)(12 101 31)(13 107 53)(14 108 54)(15 103 49)(16 104 50)(17 105 51)(18 106 52)(19 45 79)(20 46 80)(21 47 81)(22 48 82)(23 43 83)(24 44 84)(25 65 85)(26 66 86)(27 61 87)(28 62 88)(29 63 89)(30 64 90)(37 76 69)(38 77 70)(39 78 71)(40 73 72)(41 74 67)(42 75 68)
(1 78 8)(2 73 9)(3 74 10)(4 75 11)(5 76 12)(6 77 7)(13 26 24)(14 27 19)(15 28 20)(16 29 21)(17 30 22)(18 25 23)(31 95 37)(32 96 38)(33 91 39)(34 92 40)(35 93 41)(36 94 42)(43 106 65)(44 107 66)(45 108 61)(46 103 62)(47 104 63)(48 105 64)(49 88 80)(50 89 81)(51 90 82)(52 85 83)(53 86 84)(54 87 79)(55 69 101)(56 70 102)(57 71 97)(58 72 98)(59 67 99)(60 68 100)
(1 107)(2 108)(3 103)(4 104)(5 105)(6 106)(7 65)(8 66)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 55)(18 56)(19 72)(20 67)(21 68)(22 69)(23 70)(24 71)(25 102)(26 97)(27 98)(28 99)(29 100)(30 101)(31 90)(32 85)(33 86)(34 87)(35 88)(36 89)(37 82)(38 83)(39 84)(40 79)(41 80)(42 81)(43 77)(44 78)(45 73)(46 74)(47 75)(48 76)(49 93)(50 94)(51 95)(52 96)(53 91)(54 92)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C···3AO6A6B6C···6AO6AP6AQ6AR6AS
order1222333···3666···66666
size112727112···2112···227272727

90 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6
kernelC6×C33⋊C2C3×C33⋊C2C33×C6C2×C33⋊C2C33⋊C2C32×C6C32×C6C33C3×C6C32
# reps12124213132626

Matrix representation of C6×C33⋊C2 in GL6(𝔽7)

600000
060000
003000
000300
000020
000002
,
400000
020000
001000
000100
000040
000002
,
100000
010000
001000
000100
000020
000004
,
200000
040000
004000
000200
000010
000001
,
050000
300000
000100
001000
000001
000010

G:=sub<GL(6,GF(7))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6×C33⋊C2 in GAP, Magma, Sage, TeX

C_6\times C_3^3\rtimes C_2
% in TeX

G:=Group("C6xC3^3:C2");
// GroupNames label

G:=SmallGroup(324,174);
// by ID

G=gap.SmallGroup(324,174);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽