direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6.D4, C4⋊C4⋊37D6, D6.37(C2×D4), D6⋊C4⋊64C22, (C2×C6).49C24, C6.42(C22×D4), (C22×S3).95D4, (C22×D12).8C2, (C22×C4).334D6, C22.132(S3×D4), (C2×C12).616C23, Dic3⋊C4⋊51C22, C6⋊3(C22.D4), C22.83(S3×C23), (C2×D12).205C22, C22.76(C4○D12), (C22×S3).12C23, (S3×C23).99C22, (C22×C6).398C23, C23.337(C22×S3), (C22×C12).359C22, C22.35(Q8⋊3S3), (C2×Dic3).187C23, (C22×Dic3).212C22, (C6×C4⋊C4)⋊11C2, (C2×C4⋊C4)⋊14S3, C2.14(C2×S3×D4), (C2×D6⋊C4)⋊33C2, (S3×C2×C4)⋊68C22, (S3×C22×C4)⋊20C2, C6.19(C2×C4○D4), (C3×C4⋊C4)⋊45C22, C2.21(C2×C4○D12), (C2×C6).388(C2×D4), C2.6(C2×Q8⋊3S3), C3⋊3(C2×C22.D4), (C2×Dic3⋊C4)⋊23C2, (C2×C6).106(C4○D4), (C2×C4).140(C22×S3), SmallGroup(192,1064)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 952 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×10], C22, C22 [×6], C22 [×26], S3 [×6], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×22], D4 [×8], C23, C23 [×18], Dic3 [×4], C12 [×6], D6 [×4], D6 [×22], C2×C6, C2×C6 [×6], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C2×D4 [×8], C24 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×6], C2×C12 [×6], C22×S3 [×8], C22×S3 [×10], C22×C6, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4, C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×4], D6⋊C4 [×12], C3×C4⋊C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12 [×3], S3×C23 [×2], C2×C22.D4, D6.D4 [×8], C2×Dic3⋊C4, C2×D6⋊C4 [×3], C6×C4⋊C4, S3×C22×C4, C22×D12, C2×D6.D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C4○D12 [×2], S3×D4 [×2], Q8⋊3S3 [×2], S3×C23, C2×C22.D4, D6.D4 [×4], C2×C4○D12, C2×S3×D4, C2×Q8⋊3S3, C2×D6.D4
Generators and relations
G = < a,b,c,d,e | a2=b6=c2=d4=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >
►On 96 points
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 75)(20 76)(21 77)(22 78)(23 73)(24 74)(25 71)(26 72)(27 67)(28 68)(29 69)(30 70)(31 87)(32 88)(33 89)(34 90)(35 85)(36 86)(37 83)(38 84)(39 79)(40 80)(41 81)(42 82)(49 95)(50 96)(51 91)(52 92)(53 93)(54 94)
(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 51)(2 50)(3 49)(4 54)(5 53)(6 52)(7 56)(8 55)(9 60)(10 59)(11 58)(12 57)(13 46)(14 45)(15 44)(16 43)(17 48)(18 47)(19 37)(20 42)(21 41)(22 40)(23 39)(24 38)(25 31)(26 36)(27 35)(28 34)(29 33)(30 32)(61 94)(62 93)(63 92)(64 91)(65 96)(66 95)(67 85)(68 90)(69 89)(70 88)(71 87)(72 86)(73 79)(74 84)(75 83)(76 82)(77 81)(78 80)
(1 76 16 68)(2 77 17 69)(3 78 18 70)(4 73 13 71)(5 74 14 72)(6 75 15 67)(7 31 91 39)(8 32 92 40)(9 33 93 41)(10 34 94 42)(11 35 95 37)(12 36 96 38)(19 55 27 63)(20 56 28 64)(21 57 29 65)(22 58 30 66)(23 59 25 61)(24 60 26 62)(43 87 51 79)(44 88 52 80)(45 89 53 81)(46 90 54 82)(47 85 49 83)(48 86 50 84)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 92 16 95)(14 93 17 96)(15 94 18 91)(19 82 22 79)(20 83 23 80)(21 84 24 81)(25 88 28 85)(26 89 29 86)(27 90 30 87)(31 67 34 70)(32 68 35 71)(33 69 36 72)(37 73 40 76)(38 74 41 77)(39 75 42 78)(43 63 46 66)(44 64 47 61)(45 65 48 62)(49 59 52 56)(50 60 53 57)(51 55 54 58)
G:=sub<Sym(96)| (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,71)(26,72)(27,67)(28,68)(29,69)(30,70)(31,87)(32,88)(33,89)(34,90)(35,85)(36,86)(37,83)(38,84)(39,79)(40,80)(41,81)(42,82)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (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), (1,51)(2,50)(3,49)(4,54)(5,53)(6,52)(7,56)(8,55)(9,60)(10,59)(11,58)(12,57)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,37)(20,42)(21,41)(22,40)(23,39)(24,38)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32)(61,94)(62,93)(63,92)(64,91)(65,96)(66,95)(67,85)(68,90)(69,89)(70,88)(71,87)(72,86)(73,79)(74,84)(75,83)(76,82)(77,81)(78,80), (1,76,16,68)(2,77,17,69)(3,78,18,70)(4,73,13,71)(5,74,14,72)(6,75,15,67)(7,31,91,39)(8,32,92,40)(9,33,93,41)(10,34,94,42)(11,35,95,37)(12,36,96,38)(19,55,27,63)(20,56,28,64)(21,57,29,65)(22,58,30,66)(23,59,25,61)(24,60,26,62)(43,87,51,79)(44,88,52,80)(45,89,53,81)(46,90,54,82)(47,85,49,83)(48,86,50,84), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,92,16,95)(14,93,17,96)(15,94,18,91)(19,82,22,79)(20,83,23,80)(21,84,24,81)(25,88,28,85)(26,89,29,86)(27,90,30,87)(31,67,34,70)(32,68,35,71)(33,69,36,72)(37,73,40,76)(38,74,41,77)(39,75,42,78)(43,63,46,66)(44,64,47,61)(45,65,48,62)(49,59,52,56)(50,60,53,57)(51,55,54,58)>;
G:=Group( (1,64)(2,65)(3,66)(4,61)(5,62)(6,63)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,75)(20,76)(21,77)(22,78)(23,73)(24,74)(25,71)(26,72)(27,67)(28,68)(29,69)(30,70)(31,87)(32,88)(33,89)(34,90)(35,85)(36,86)(37,83)(38,84)(39,79)(40,80)(41,81)(42,82)(49,95)(50,96)(51,91)(52,92)(53,93)(54,94), (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), (1,51)(2,50)(3,49)(4,54)(5,53)(6,52)(7,56)(8,55)(9,60)(10,59)(11,58)(12,57)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,37)(20,42)(21,41)(22,40)(23,39)(24,38)(25,31)(26,36)(27,35)(28,34)(29,33)(30,32)(61,94)(62,93)(63,92)(64,91)(65,96)(66,95)(67,85)(68,90)(69,89)(70,88)(71,87)(72,86)(73,79)(74,84)(75,83)(76,82)(77,81)(78,80), (1,76,16,68)(2,77,17,69)(3,78,18,70)(4,73,13,71)(5,74,14,72)(6,75,15,67)(7,31,91,39)(8,32,92,40)(9,33,93,41)(10,34,94,42)(11,35,95,37)(12,36,96,38)(19,55,27,63)(20,56,28,64)(21,57,29,65)(22,58,30,66)(23,59,25,61)(24,60,26,62)(43,87,51,79)(44,88,52,80)(45,89,53,81)(46,90,54,82)(47,85,49,83)(48,86,50,84), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,92,16,95)(14,93,17,96)(15,94,18,91)(19,82,22,79)(20,83,23,80)(21,84,24,81)(25,88,28,85)(26,89,29,86)(27,90,30,87)(31,67,34,70)(32,68,35,71)(33,69,36,72)(37,73,40,76)(38,74,41,77)(39,75,42,78)(43,63,46,66)(44,64,47,61)(45,65,48,62)(49,59,52,56)(50,60,53,57)(51,55,54,58) );
G=PermutationGroup([(1,64),(2,65),(3,66),(4,61),(5,62),(6,63),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,59),(14,60),(15,55),(16,56),(17,57),(18,58),(19,75),(20,76),(21,77),(22,78),(23,73),(24,74),(25,71),(26,72),(27,67),(28,68),(29,69),(30,70),(31,87),(32,88),(33,89),(34,90),(35,85),(36,86),(37,83),(38,84),(39,79),(40,80),(41,81),(42,82),(49,95),(50,96),(51,91),(52,92),(53,93),(54,94)], [(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)], [(1,51),(2,50),(3,49),(4,54),(5,53),(6,52),(7,56),(8,55),(9,60),(10,59),(11,58),(12,57),(13,46),(14,45),(15,44),(16,43),(17,48),(18,47),(19,37),(20,42),(21,41),(22,40),(23,39),(24,38),(25,31),(26,36),(27,35),(28,34),(29,33),(30,32),(61,94),(62,93),(63,92),(64,91),(65,96),(66,95),(67,85),(68,90),(69,89),(70,88),(71,87),(72,86),(73,79),(74,84),(75,83),(76,82),(77,81),(78,80)], [(1,76,16,68),(2,77,17,69),(3,78,18,70),(4,73,13,71),(5,74,14,72),(6,75,15,67),(7,31,91,39),(8,32,92,40),(9,33,93,41),(10,34,94,42),(11,35,95,37),(12,36,96,38),(19,55,27,63),(20,56,28,64),(21,57,29,65),(22,58,30,66),(23,59,25,61),(24,60,26,62),(43,87,51,79),(44,88,52,80),(45,89,53,81),(46,90,54,82),(47,85,49,83),(48,86,50,84)], [(1,11,4,8),(2,12,5,9),(3,7,6,10),(13,92,16,95),(14,93,17,96),(15,94,18,91),(19,82,22,79),(20,83,23,80),(21,84,24,81),(25,88,28,85),(26,89,29,86),(27,90,30,87),(31,67,34,70),(32,68,35,71),(33,69,36,72),(37,73,40,76),(38,74,41,77),(39,75,42,78),(43,63,46,66),(44,64,47,61),(45,65,48,62),(49,59,52,56),(50,60,53,57),(51,55,54,58)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[8,12,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[1,8,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 | Q8⋊3S3 |
| kernel | C2×D6.D4 | D6.D4 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C4⋊C4 | S3×C22×C4 | C22×D12 | C2×C4⋊C4 | C22×S3 | C4⋊C4 | C22×C4 | C2×C6 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times D_6.D_4
% in TeX
G:=Group("C2xD6.D4"); // GroupNames label
G:=SmallGroup(192,1064);
// by ID
G=gap.SmallGroup(192,1064);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^4=1,e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations