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## G = C2×C6×3- 1+2order 324 = 22·34

### Direct product of C2×C6 and 3- 1+2

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C6×3- 1+2
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — C6×3- 1+2 — C2×C6×3- 1+2
 Lower central C1 — C3 — C2×C6×3- 1+2
 Upper central C1 — C62 — C2×C6×3- 1+2

Generators and relations for C2×C6×3- 1+2
G = < a,b,c,d | a2=b6=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 250 in 190 conjugacy classes, 160 normal (10 characteristic)
C1, C2, C3, C3, C3, C22, C6, C6, C9, C32, C32, C32, C2×C6, C2×C6, C2×C6, C18, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C2×C18, C62, C62, C62, C3×C18, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×C18, C22×3- 1+2, C3×C62, C6×3- 1+2, C2×C6×3- 1+2
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, 3- 1+2, C33, C62, C2×3- 1+2, C32×C6, C3×3- 1+2, C22×3- 1+2, C3×C62, C6×3- 1+2, C2×C6×3- 1+2

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

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

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

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

132 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3N 6A ··· 6X 6Y ··· 6AP 9A ··· 9R 18A ··· 18BB order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 3- 1+2 C2×3- 1+2 kernel C2×C6×3- 1+2 C6×3- 1+2 C6×C18 C22×3- 1+2 C3×C62 C3×C18 C2×3- 1+2 C32×C6 C2×C6 C6 # reps 1 3 6 18 2 18 54 6 6 18

Matrix representation of C2×C6×3- 1+2 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 7 0 0 0 0 7 0 12 0 0 0 11 0 6 8 12
,
 1 0 0 0 0 1 12 7 0 0 11 0 0 0 0 7
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[7,0,0,0,0,7,0,6,0,0,0,8,0,12,11,12],[1,0,0,0,0,1,0,0,0,12,11,0,0,7,0,7] >;

C2×C6×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times C_6\times 3_-^{1+2}
% in TeX

G:=Group("C2xC6xES-(3,1)");
// GroupNames label

G:=SmallGroup(324,153);
// by ID

G=gap.SmallGroup(324,153);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,735]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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