direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C42⋊3S3, C42⋊38D6, (C2×C42)⋊5S3, (C4×C12)⋊47C22, (C2×C6).22C24, C6⋊1(C42⋊2C2), D6⋊C4.81C22, (C22×C4).425D6, (C2×C12).695C23, Dic3⋊C4⋊41C22, (C22×S3).4C23, C22.65(S3×C23), (C2×Dic3).6C23, C22.70(C4○D12), (S3×C23).29C22, C23.327(C22×S3), (C22×C6).384C23, (C22×C12).505C22, (C22×Dic3).75C22, (C2×C4×C12)⋊4C2, C6.9(C2×C4○D4), C3⋊1(C2×C42⋊2C2), (C2×D6⋊C4).16C2, C2.11(C2×C4○D12), (C2×Dic3⋊C4)⋊16C2, (C2×C6).98(C4○D4), (C2×C4).650(C22×S3), SmallGroup(192,1037)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C42⋊3S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c-1, ede=d-1 >
Subgroups: 600 in 246 conjugacy classes, 111 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, Dic3⋊C4, D6⋊C4, C4×C12, C22×Dic3, C22×C12, S3×C23, C2×C42⋊2C2, C42⋊3S3, C2×Dic3⋊C4, C2×D6⋊C4, C2×C4×C12, C2×C42⋊3S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C42⋊2C2, C2×C4○D4, C4○D12, S3×C23, C2×C42⋊2C2, C42⋊3S3, C2×C4○D12, C2×C42⋊3S3
►On 96 points
Generators in S96
(1 72)(2 69)(3 70)(4 71)(5 67)(6 68)(7 65)(8 66)(9 17)(10 18)(11 19)(12 20)(13 44)(14 41)(15 42)(16 43)(21 51)(22 52)(23 49)(24 50)(25 40)(26 37)(27 38)(28 39)(29 94)(30 95)(31 96)(32 93)(33 54)(34 55)(35 56)(36 53)(45 74)(46 75)(47 76)(48 73)(57 83)(58 84)(59 81)(60 82)(61 89)(62 90)(63 91)(64 92)(77 88)(78 85)(79 86)(80 87)
(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 23 39 58)(2 24 40 59)(3 21 37 60)(4 22 38 57)(5 20 48 63)(6 17 45 64)(7 18 46 61)(8 19 47 62)(9 74 92 68)(10 75 89 65)(11 76 90 66)(12 73 91 67)(13 53 30 86)(14 54 31 87)(15 55 32 88)(16 56 29 85)(25 81 69 50)(26 82 70 51)(27 83 71 52)(28 84 72 49)(33 96 80 41)(34 93 77 42)(35 94 78 43)(36 95 79 44)
(1 54 11)(2 55 12)(3 56 9)(4 53 10)(5 81 42)(6 82 43)(7 83 44)(8 84 41)(13 65 57)(14 66 58)(15 67 59)(16 68 60)(17 70 35)(18 71 36)(19 72 33)(20 69 34)(21 29 74)(22 30 75)(23 31 76)(24 32 73)(25 77 63)(26 78 64)(27 79 61)(28 80 62)(37 85 92)(38 86 89)(39 87 90)(40 88 91)(45 51 94)(46 52 95)(47 49 96)(48 50 93)
(2 40)(4 38)(5 44)(6 96)(7 42)(8 94)(9 56)(10 86)(11 54)(12 88)(13 67)(14 74)(15 65)(16 76)(17 35)(18 79)(19 33)(20 77)(21 58)(22 24)(23 60)(25 69)(27 71)(29 66)(30 73)(31 68)(32 75)(34 63)(36 61)(41 45)(43 47)(46 93)(48 95)(49 82)(50 52)(51 84)(53 89)(55 91)(57 59)(62 80)(64 78)(81 83)(85 92)(87 90)
G:=sub<Sym(96)| (1,72)(2,69)(3,70)(4,71)(5,67)(6,68)(7,65)(8,66)(9,17)(10,18)(11,19)(12,20)(13,44)(14,41)(15,42)(16,43)(21,51)(22,52)(23,49)(24,50)(25,40)(26,37)(27,38)(28,39)(29,94)(30,95)(31,96)(32,93)(33,54)(34,55)(35,56)(36,53)(45,74)(46,75)(47,76)(48,73)(57,83)(58,84)(59,81)(60,82)(61,89)(62,90)(63,91)(64,92)(77,88)(78,85)(79,86)(80,87), (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), (1,23,39,58)(2,24,40,59)(3,21,37,60)(4,22,38,57)(5,20,48,63)(6,17,45,64)(7,18,46,61)(8,19,47,62)(9,74,92,68)(10,75,89,65)(11,76,90,66)(12,73,91,67)(13,53,30,86)(14,54,31,87)(15,55,32,88)(16,56,29,85)(25,81,69,50)(26,82,70,51)(27,83,71,52)(28,84,72,49)(33,96,80,41)(34,93,77,42)(35,94,78,43)(36,95,79,44), (1,54,11)(2,55,12)(3,56,9)(4,53,10)(5,81,42)(6,82,43)(7,83,44)(8,84,41)(13,65,57)(14,66,58)(15,67,59)(16,68,60)(17,70,35)(18,71,36)(19,72,33)(20,69,34)(21,29,74)(22,30,75)(23,31,76)(24,32,73)(25,77,63)(26,78,64)(27,79,61)(28,80,62)(37,85,92)(38,86,89)(39,87,90)(40,88,91)(45,51,94)(46,52,95)(47,49,96)(48,50,93), (2,40)(4,38)(5,44)(6,96)(7,42)(8,94)(9,56)(10,86)(11,54)(12,88)(13,67)(14,74)(15,65)(16,76)(17,35)(18,79)(19,33)(20,77)(21,58)(22,24)(23,60)(25,69)(27,71)(29,66)(30,73)(31,68)(32,75)(34,63)(36,61)(41,45)(43,47)(46,93)(48,95)(49,82)(50,52)(51,84)(53,89)(55,91)(57,59)(62,80)(64,78)(81,83)(85,92)(87,90)>;
G:=Group( (1,72)(2,69)(3,70)(4,71)(5,67)(6,68)(7,65)(8,66)(9,17)(10,18)(11,19)(12,20)(13,44)(14,41)(15,42)(16,43)(21,51)(22,52)(23,49)(24,50)(25,40)(26,37)(27,38)(28,39)(29,94)(30,95)(31,96)(32,93)(33,54)(34,55)(35,56)(36,53)(45,74)(46,75)(47,76)(48,73)(57,83)(58,84)(59,81)(60,82)(61,89)(62,90)(63,91)(64,92)(77,88)(78,85)(79,86)(80,87), (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), (1,23,39,58)(2,24,40,59)(3,21,37,60)(4,22,38,57)(5,20,48,63)(6,17,45,64)(7,18,46,61)(8,19,47,62)(9,74,92,68)(10,75,89,65)(11,76,90,66)(12,73,91,67)(13,53,30,86)(14,54,31,87)(15,55,32,88)(16,56,29,85)(25,81,69,50)(26,82,70,51)(27,83,71,52)(28,84,72,49)(33,96,80,41)(34,93,77,42)(35,94,78,43)(36,95,79,44), (1,54,11)(2,55,12)(3,56,9)(4,53,10)(5,81,42)(6,82,43)(7,83,44)(8,84,41)(13,65,57)(14,66,58)(15,67,59)(16,68,60)(17,70,35)(18,71,36)(19,72,33)(20,69,34)(21,29,74)(22,30,75)(23,31,76)(24,32,73)(25,77,63)(26,78,64)(27,79,61)(28,80,62)(37,85,92)(38,86,89)(39,87,90)(40,88,91)(45,51,94)(46,52,95)(47,49,96)(48,50,93), (2,40)(4,38)(5,44)(6,96)(7,42)(8,94)(9,56)(10,86)(11,54)(12,88)(13,67)(14,74)(15,65)(16,76)(17,35)(18,79)(19,33)(20,77)(21,58)(22,24)(23,60)(25,69)(27,71)(29,66)(30,73)(31,68)(32,75)(34,63)(36,61)(41,45)(43,47)(46,93)(48,95)(49,82)(50,52)(51,84)(53,89)(55,91)(57,59)(62,80)(64,78)(81,83)(85,92)(87,90) );
G=PermutationGroup([[(1,72),(2,69),(3,70),(4,71),(5,67),(6,68),(7,65),(8,66),(9,17),(10,18),(11,19),(12,20),(13,44),(14,41),(15,42),(16,43),(21,51),(22,52),(23,49),(24,50),(25,40),(26,37),(27,38),(28,39),(29,94),(30,95),(31,96),(32,93),(33,54),(34,55),(35,56),(36,53),(45,74),(46,75),(47,76),(48,73),(57,83),(58,84),(59,81),(60,82),(61,89),(62,90),(63,91),(64,92),(77,88),(78,85),(79,86),(80,87)], [(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)], [(1,23,39,58),(2,24,40,59),(3,21,37,60),(4,22,38,57),(5,20,48,63),(6,17,45,64),(7,18,46,61),(8,19,47,62),(9,74,92,68),(10,75,89,65),(11,76,90,66),(12,73,91,67),(13,53,30,86),(14,54,31,87),(15,55,32,88),(16,56,29,85),(25,81,69,50),(26,82,70,51),(27,83,71,52),(28,84,72,49),(33,96,80,41),(34,93,77,42),(35,94,78,43),(36,95,79,44)], [(1,54,11),(2,55,12),(3,56,9),(4,53,10),(5,81,42),(6,82,43),(7,83,44),(8,84,41),(13,65,57),(14,66,58),(15,67,59),(16,68,60),(17,70,35),(18,71,36),(19,72,33),(20,69,34),(21,29,74),(22,30,75),(23,31,76),(24,32,73),(25,77,63),(26,78,64),(27,79,61),(28,80,62),(37,85,92),(38,86,89),(39,87,90),(40,88,91),(45,51,94),(46,52,95),(47,49,96),(48,50,93)], [(2,40),(4,38),(5,44),(6,96),(7,42),(8,94),(9,56),(10,86),(11,54),(12,88),(13,67),(14,74),(15,65),(16,76),(17,35),(18,79),(19,33),(20,77),(21,58),(22,24),(23,60),(25,69),(27,71),(29,66),(30,73),(31,68),(32,75),(34,63),(36,61),(41,45),(43,47),(46,93),(48,95),(49,82),(50,52),(51,84),(53,89),(55,91),(57,59),(62,80),(64,78),(81,83),(85,92),(87,90)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | ··· | 4L | 4M | ··· | 4R | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C2×C42⋊3S3 | C42⋊3S3 | C2×Dic3⋊C4 | C2×D6⋊C4 | C2×C4×C12 | C2×C42 | C42 | C22×C4 | C2×C6 | C22 |
| # reps | 1 | 8 | 3 | 3 | 1 | 1 | 4 | 3 | 12 | 24 |
Matrix representation of C2×C42⋊3S3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 2 | 4 | 0 | 0 |
| 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 9 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,2,9,0,0,0,4,11,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,2,9,0,0,0,4,11],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,1,12],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12] >;
C2×C42⋊3S3 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_3S_3
% in TeX
G:=Group("C2xC4^2:3S3"); // GroupNames label
G:=SmallGroup(192,1037);
// by ID
G=gap.SmallGroup(192,1037);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=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*c^2,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=d^-1>;
// generators/relations