direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C22×C6, C62.273C23, (C2×C62)⋊10C4, C3⋊2(C23×C12), C6⋊2(C22×C12), C62⋊25(C2×C4), (C22×C6)⋊7C12, C24.4(C3×S3), C32⋊7(C23×C4), (C3×C6).51C24, (C23×C6).15S3, C23.46(S3×C6), C6.82(S3×C23), (C23×C6).13C6, C6.14(C23×C6), (C22×C6).177D6, (C22×C62).4C2, (C2×C62).119C22, (C2×C6)⋊12(C2×C12), C2.2(S3×C22×C6), (C3×C6)⋊7(C22×C4), C22.33(S3×C2×C6), (C22×C6).72(C2×C6), (C2×C6).74(C22×C6), (C2×C6).350(C22×S3), SmallGroup(288,1001)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C22×C6 |
Generators and relations for Dic3×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 810 in 539 conjugacy classes, 370 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C24, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C23×C4, C3×Dic3, C62, C22×Dic3, C22×C12, C23×C6, C23×C6, C6×Dic3, C2×C62, C23×Dic3, C23×C12, Dic3×C2×C6, C22×C62, Dic3×C22×C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C24, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C23×C4, C3×Dic3, S3×C6, C22×Dic3, C22×C12, S3×C23, C23×C6, C6×Dic3, S3×C2×C6, C23×Dic3, C23×C12, Dic3×C2×C6, S3×C22×C6, Dic3×C22×C6
►On 96 points
Generators in S96
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 35)(14 36)(15 31)(16 32)(17 33)(18 34)(19 51)(20 52)(21 53)(22 54)(23 49)(24 50)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)(55 87)(56 88)(57 89)(58 90)(59 85)(60 86)(61 83)(62 84)(63 79)(64 80)(65 81)(66 82)(73 95)(74 96)(75 91)(76 92)(77 93)(78 94)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 52)(38 53)(39 54)(40 49)(41 50)(42 51)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(61 76)(62 77)(63 78)(64 73)(65 74)(66 75)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(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)
(1 14 3 16 5 18)(2 15 4 17 6 13)(7 93 11 91 9 95)(8 94 12 92 10 96)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 37 33 39 35 41)(32 38 34 40 36 42)(43 49 45 51 47 53)(44 50 46 52 48 54)(55 65 59 63 57 61)(56 66 60 64 58 62)(67 77 71 75 69 73)(68 78 72 76 70 74)(79 89 83 87 81 85)(80 90 84 88 82 86)
(1 61 16 59)(2 62 17 60)(3 63 18 55)(4 64 13 56)(5 65 14 57)(6 66 15 58)(7 54 91 46)(8 49 92 47)(9 50 93 48)(10 51 94 43)(11 52 95 44)(12 53 96 45)(19 78 27 70)(20 73 28 71)(21 74 29 72)(22 75 30 67)(23 76 25 68)(24 77 26 69)(31 90 39 82)(32 85 40 83)(33 86 41 84)(34 87 42 79)(35 88 37 80)(36 89 38 81)
G:=sub<Sym(96)| (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,35)(14,36)(15,31)(16,32)(17,33)(18,34)(19,51)(20,52)(21,53)(22,54)(23,49)(24,50)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46)(55,87)(56,88)(57,89)(58,90)(59,85)(60,86)(61,83)(62,84)(63,79)(64,80)(65,81)(66,82)(73,95)(74,96)(75,91)(76,92)(77,93)(78,94), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,52)(38,53)(39,54)(40,49)(41,50)(42,51)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(61,76)(62,77)(63,78)(64,73)(65,74)(66,75)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (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), (1,14,3,16,5,18)(2,15,4,17,6,13)(7,93,11,91,9,95)(8,94,12,92,10,96)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,37,33,39,35,41)(32,38,34,40,36,42)(43,49,45,51,47,53)(44,50,46,52,48,54)(55,65,59,63,57,61)(56,66,60,64,58,62)(67,77,71,75,69,73)(68,78,72,76,70,74)(79,89,83,87,81,85)(80,90,84,88,82,86), (1,61,16,59)(2,62,17,60)(3,63,18,55)(4,64,13,56)(5,65,14,57)(6,66,15,58)(7,54,91,46)(8,49,92,47)(9,50,93,48)(10,51,94,43)(11,52,95,44)(12,53,96,45)(19,78,27,70)(20,73,28,71)(21,74,29,72)(22,75,30,67)(23,76,25,68)(24,77,26,69)(31,90,39,82)(32,85,40,83)(33,86,41,84)(34,87,42,79)(35,88,37,80)(36,89,38,81)>;
G:=Group( (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,35)(14,36)(15,31)(16,32)(17,33)(18,34)(19,51)(20,52)(21,53)(22,54)(23,49)(24,50)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46)(55,87)(56,88)(57,89)(58,90)(59,85)(60,86)(61,83)(62,84)(63,79)(64,80)(65,81)(66,82)(73,95)(74,96)(75,91)(76,92)(77,93)(78,94), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,52)(38,53)(39,54)(40,49)(41,50)(42,51)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(61,76)(62,77)(63,78)(64,73)(65,74)(66,75)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (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), (1,14,3,16,5,18)(2,15,4,17,6,13)(7,93,11,91,9,95)(8,94,12,92,10,96)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,37,33,39,35,41)(32,38,34,40,36,42)(43,49,45,51,47,53)(44,50,46,52,48,54)(55,65,59,63,57,61)(56,66,60,64,58,62)(67,77,71,75,69,73)(68,78,72,76,70,74)(79,89,83,87,81,85)(80,90,84,88,82,86), (1,61,16,59)(2,62,17,60)(3,63,18,55)(4,64,13,56)(5,65,14,57)(6,66,15,58)(7,54,91,46)(8,49,92,47)(9,50,93,48)(10,51,94,43)(11,52,95,44)(12,53,96,45)(19,78,27,70)(20,73,28,71)(21,74,29,72)(22,75,30,67)(23,76,25,68)(24,77,26,69)(31,90,39,82)(32,85,40,83)(33,86,41,84)(34,87,42,79)(35,88,37,80)(36,89,38,81) );
G=PermutationGroup([[(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,35),(14,36),(15,31),(16,32),(17,33),(18,34),(19,51),(20,52),(21,53),(22,54),(23,49),(24,50),(25,47),(26,48),(27,43),(28,44),(29,45),(30,46),(55,87),(56,88),(57,89),(58,90),(59,85),(60,86),(61,83),(62,84),(63,79),(64,80),(65,81),(66,82),(73,95),(74,96),(75,91),(76,92),(77,93),(78,94)], [(1,23),(2,24),(3,19),(4,20),(5,21),(6,22),(7,90),(8,85),(9,86),(10,87),(11,88),(12,89),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(37,52),(38,53),(39,54),(40,49),(41,50),(42,51),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(61,76),(62,77),(63,78),(64,73),(65,74),(66,75),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)], [(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)], [(1,14,3,16,5,18),(2,15,4,17,6,13),(7,93,11,91,9,95),(8,94,12,92,10,96),(19,25,21,27,23,29),(20,26,22,28,24,30),(31,37,33,39,35,41),(32,38,34,40,36,42),(43,49,45,51,47,53),(44,50,46,52,48,54),(55,65,59,63,57,61),(56,66,60,64,58,62),(67,77,71,75,69,73),(68,78,72,76,70,74),(79,89,83,87,81,85),(80,90,84,88,82,86)], [(1,61,16,59),(2,62,17,60),(3,63,18,55),(4,64,13,56),(5,65,14,57),(6,66,15,58),(7,54,91,46),(8,49,92,47),(9,50,93,48),(10,51,94,43),(11,52,95,44),(12,53,96,45),(19,78,27,70),(20,73,28,71),(21,74,29,72),(22,75,30,67),(23,76,25,68),(24,77,26,69),(31,90,39,82),(32,85,40,83),(33,86,41,84),(34,87,42,79),(35,88,37,80),(36,89,38,81)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4P | 6A | ··· | 6AD | 6AE | ··· | 6BW | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 |
| kernel | Dic3×C22×C6 | Dic3×C2×C6 | C22×C62 | C23×Dic3 | C2×C62 | C22×Dic3 | C23×C6 | C22×C6 | C23×C6 | C22×C6 | C22×C6 | C24 | C23 | C23 |
| # reps | 1 | 14 | 1 | 2 | 16 | 28 | 2 | 32 | 1 | 8 | 7 | 2 | 16 | 14 |
Matrix representation of Dic3×C22×C6 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,0,4,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,3],[12,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,1,0] >;
Dic3×C22×C6 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2^2\times C_6 % in TeX
G:=Group("Dic3xC2^2xC6"); // GroupNames label
G:=SmallGroup(288,1001);
// by ID
G=gap.SmallGroup(288,1001);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations