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G = C12×He3order 324 = 22·34

Direct product of C12 and He3

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

Aliases: C12×He3, C338C12, C12.1C33, C2.(C6×He3), (C3×C12)⋊C32, (C32×C12)⋊2C3, C323(C3×C12), (C6×He3).6C2, C6.2(C32×C6), C6.12(C2×He3), (C2×He3).14C6, (C32×C6).12C6, C3.1(C32×C12), (C3×C6).10(C3×C6), SmallGroup(324,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C12×He3
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C6×He3 — C12×He3
 Lower central C1 — C3 — C12×He3
 Upper central C1 — C3×C12 — C12×He3

Generators and relations for C12×He3
G = < a,b,c,d | a12=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 312 in 168 conjugacy classes, 96 normal (12 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, C12, C12, C12, C3×C6, C3×C6, C3×C6, He3, C33, C3×C12, C3×C12, C3×C12, C2×He3, C32×C6, C3×He3, C4×He3, C32×C12, C6×He3, C12×He3
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, He3, C33, C3×C12, C2×He3, C32×C6, C3×He3, C4×He3, C32×C12, C6×He3, C12×He3

Smallest permutation representation of C12×He3
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 29 55)(2 30 56)(3 31 57)(4 32 58)(5 33 59)(6 34 60)(7 35 49)(8 36 50)(9 25 51)(10 26 52)(11 27 53)(12 28 54)(13 83 37)(14 84 38)(15 73 39)(16 74 40)(17 75 41)(18 76 42)(19 77 43)(20 78 44)(21 79 45)(22 80 46)(23 81 47)(24 82 48)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(85 89 93)(86 90 94)(87 91 95)(88 92 96)(97 101 105)(98 102 106)(99 103 107)(100 104 108)
(1 25 59)(2 26 60)(3 27 49)(4 28 50)(5 29 51)(6 30 52)(7 31 53)(8 32 54)(9 33 55)(10 34 56)(11 35 57)(12 36 58)(13 41 79)(14 42 80)(15 43 81)(16 44 82)(17 45 83)(18 46 84)(19 47 73)(20 48 74)(21 37 75)(22 38 76)(23 39 77)(24 40 78)(61 103 88)(62 104 89)(63 105 90)(64 106 91)(65 107 92)(66 108 93)(67 97 94)(68 98 95)(69 99 96)(70 100 85)(71 101 86)(72 102 87)
(1 86 84)(2 87 73)(3 88 74)(4 89 75)(5 90 76)(6 91 77)(7 92 78)(8 93 79)(9 94 80)(10 95 81)(11 96 82)(12 85 83)(13 32 66)(14 33 67)(15 34 68)(16 35 69)(17 36 70)(18 25 71)(19 26 72)(20 27 61)(21 28 62)(22 29 63)(23 30 64)(24 31 65)(37 50 104)(38 51 105)(39 52 106)(40 53 107)(41 54 108)(42 55 97)(43 56 98)(44 57 99)(45 58 100)(46 59 101)(47 60 102)(48 49 103)

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

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

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

132 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 4A 4B 6A ··· 6H 6I ··· 6AF 12A ··· 12P 12Q ··· 12BL order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 ··· 1 3 ··· 3 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 He3 C2×He3 C4×He3 kernel C12×He3 C6×He3 C4×He3 C32×C12 C3×He3 C2×He3 C32×C6 He3 C33 C12 C6 C3 # reps 1 1 18 8 2 18 8 36 16 6 6 12

Matrix representation of C12×He3 in GL4(𝔽13) generated by

 9 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 3 0 0 0 0 3 12 5 0 0 9 0 0 0 0 1
,
 1 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 3 0 0 0 0 5 5 12 0 0 0 3 0 11 4 8
G:=sub<GL(4,GF(13))| [9,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[3,0,0,0,0,3,0,0,0,12,9,0,0,5,0,1],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,5,0,11,0,5,0,4,0,12,3,8] >;

C12×He3 in GAP, Magma, Sage, TeX

C_{12}\times {\rm He}_3
% in TeX

G:=Group("C12xHe3");
// GroupNames label

G:=SmallGroup(324,106);
// by ID

G=gap.SmallGroup(324,106);
# by ID

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

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

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