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## G = C12×3- 1+2order 324 = 22·34

### Direct product of C12 and 3- 1+2

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C12×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C6×3- 1+2 — C12×3- 1+2
 Lower central C1 — C3 — C12×3- 1+2
 Upper central C1 — C3×C12 — C12×3- 1+2

Generators and relations for C12×3- 1+2
G = < a,b,c | a12=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 150 in 114 conjugacy classes, 96 normal (15 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C32, C32, C32, C12, C12, C12, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C36, C3×C12, C3×C12, C3×C12, C3×C18, C2×3- 1+2, C32×C6, C3×3- 1+2, C3×C36, C4×3- 1+2, C32×C12, C6×3- 1+2, C12×3- 1+2
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, 3- 1+2, C33, C3×C12, C2×3- 1+2, C32×C6, C3×3- 1+2, C4×3- 1+2, C32×C12, C6×3- 1+2, C12×3- 1+2

Smallest permutation representation of C12×3- 1+2
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 45 21 93 57 73 25 61 98)(2 46 22 94 58 74 26 62 99)(3 47 23 95 59 75 27 63 100)(4 48 24 96 60 76 28 64 101)(5 37 13 85 49 77 29 65 102)(6 38 14 86 50 78 30 66 103)(7 39 15 87 51 79 31 67 104)(8 40 16 88 52 80 32 68 105)(9 41 17 89 53 81 33 69 106)(10 42 18 90 54 82 34 70 107)(11 43 19 91 55 83 35 71 108)(12 44 20 92 56 84 36 72 97)
(1 85 33)(2 86 34)(3 87 35)(4 88 36)(5 89 25)(6 90 26)(7 91 27)(8 92 28)(9 93 29)(10 94 30)(11 95 31)(12 96 32)(13 106 73)(14 107 74)(15 108 75)(16 97 76)(17 98 77)(18 99 78)(19 100 79)(20 101 80)(21 102 81)(22 103 82)(23 104 83)(24 105 84)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)

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

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

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

132 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 4A 4B 6A ··· 6H 6I ··· 6N 9A ··· 9R 12A ··· 12P 12Q ··· 12AB 18A ··· 18R 36A ··· 36AJ order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 ··· 1 3 ··· 3 1 1 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C3 C3 C4 C6 C6 C6 C12 C12 C12 3- 1+2 C2×3- 1+2 C4×3- 1+2 kernel C12×3- 1+2 C6×3- 1+2 C3×C36 C4×3- 1+2 C32×C12 C3×3- 1+2 C3×C18 C2×3- 1+2 C32×C6 C3×C9 3- 1+2 C33 C12 C6 C3 # reps 1 1 6 18 2 2 6 18 2 12 36 4 6 6 12

Matrix representation of C12×3- 1+2 in GL4(𝔽37) generated by

 10 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31
,
 10 0 0 0 0 0 10 0 0 0 0 1 0 10 0 0
,
 26 0 0 0 0 26 0 0 0 0 10 0 0 0 0 1
G:=sub<GL(4,GF(37))| [10,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[10,0,0,0,0,0,0,10,0,10,0,0,0,0,1,0],[26,0,0,0,0,26,0,0,0,0,10,0,0,0,0,1] >;

C12×3- 1+2 in GAP, Magma, Sage, TeX

C_{12}\times 3_-^{1+2}
% in TeX

G:=Group("C12xES-(3,1)");
// GroupNames label

G:=SmallGroup(324,107);
// by ID

G=gap.SmallGroup(324,107);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,655,1034]);
// Polycyclic

G:=Group<a,b,c|a^12=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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