direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3⋊5D4, C6⋊3(C4×D4), C4⋊C4⋊52D6, (C2×D12)⋊17C4, D12⋊22(C2×C4), C12⋊3(C22×C4), D6⋊3(C22×C4), D6⋊C4⋊59C22, (C2×Dic3)⋊24D4, Dic3⋊10(C2×D4), C6.13(C23×C4), (C2×C6).47C24, C6.41(C22×D4), C22.131(S3×D4), (C22×C4).333D6, (C2×C12).579C23, (C4×Dic3)⋊64C22, (C22×D12).16C2, C22.23(S3×C23), (C2×D12).252C22, (S3×C23).98C22, C23.336(C22×S3), (C22×C6).396C23, (C22×S3).154C23, (C22×C12).215C22, C22.34(Q8⋊3S3), (C2×Dic3).305C23, (C22×Dic3).211C22, C3⋊3(C2×C4×D4), C4⋊2(S3×C2×C4), (C6×C4⋊C4)⋊9C2, C2.4(C2×S3×D4), (C2×C4)⋊9(C4×S3), (C2×C4⋊C4)⋊26S3, (C2×C12)⋊8(C2×C4), (C2×C4×Dic3)⋊5C2, (C2×D6⋊C4)⋊32C2, (S3×C2×C4)⋊67C22, (S3×C22×C4)⋊19C2, C22.73(S3×C2×C4), C2.15(S3×C22×C4), (C3×C4⋊C4)⋊44C22, C6.108(C2×C4○D4), (C2×C6).387(C2×D4), C2.2(C2×Q8⋊3S3), (C22×S3)⋊11(C2×C4), (C2×C6).196(C4○D4), (C2×C4).266(C22×S3), (C2×C6).152(C22×C4), SmallGroup(192,1062)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1048 in 426 conjugacy classes, 175 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×10], C22, C22 [×6], C22 [×32], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×30], D4 [×16], C23, C23 [×20], Dic3 [×4], Dic3 [×2], C12 [×4], C12 [×4], D6 [×8], D6 [×24], C2×C6, C2×C6 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×18], C2×D4 [×12], C24 [×2], C4×S3 [×16], D12 [×16], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×10], C2×C12 [×4], C22×S3 [×12], C22×S3 [×8], C22×C6, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×Dic3 [×4], D6⋊C4 [×8], C3×C4⋊C4 [×4], S3×C2×C4 [×8], S3×C2×C4 [×8], C2×D12 [×12], C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23 [×2], C2×C4×D4, Dic3⋊5D4 [×8], C2×C4×Dic3, C2×D6⋊C4 [×2], C6×C4⋊C4, S3×C22×C4 [×2], C22×D12, C2×Dic3⋊5D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×4], C23 [×15], D6 [×7], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×S3 [×4], C22×S3 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, S3×C2×C4 [×6], S3×D4 [×2], Q8⋊3S3 [×2], S3×C23, C2×C4×D4, Dic3⋊5D4 [×4], S3×C22×C4, C2×S3×D4, C2×Q8⋊3S3, C2×Dic3⋊5D4
Generators and relations
G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
►On 96 points
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 74)(20 75)(21 76)(22 77)(23 78)(24 73)(25 72)(26 67)(27 68)(28 69)(29 70)(30 71)(31 86)(32 87)(33 88)(34 89)(35 90)(36 85)(37 84)(38 79)(39 80)(40 81)(41 82)(42 83)(49 96)(50 91)(51 92)(52 93)(53 94)(54 95)
(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 12 4 9)(2 11 5 8)(3 10 6 7)(13 95 16 92)(14 94 17 91)(15 93 18 96)(19 79 22 82)(20 84 23 81)(21 83 24 80)(25 87 28 90)(26 86 29 89)(27 85 30 88)(31 70 34 67)(32 69 35 72)(33 68 36 71)(37 78 40 75)(38 77 41 74)(39 76 42 73)(43 62 46 65)(44 61 47 64)(45 66 48 63)(49 56 52 59)(50 55 53 58)(51 60 54 57)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 75 92 68)(8 76 93 69)(9 77 94 70)(10 78 95 71)(11 73 96 72)(12 74 91 67)(19 50 26 43)(20 51 27 44)(21 52 28 45)(22 53 29 46)(23 54 30 47)(24 49 25 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 84)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(20 24)(21 23)(25 27)(28 30)(32 36)(33 35)(37 39)(40 42)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(68 72)(69 71)(73 75)(76 78)(80 84)(81 83)(85 87)(88 90)
G:=sub<Sym(96)| (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95), (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), (1,12,4,9)(2,11,5,8)(3,10,6,7)(13,95,16,92)(14,94,17,91)(15,93,18,96)(19,79,22,82)(20,84,23,81)(21,83,24,80)(25,87,28,90)(26,86,29,89)(27,85,30,88)(31,70,34,67)(32,69,35,72)(33,68,36,71)(37,78,40,75)(38,77,41,74)(39,76,42,73)(43,62,46,65)(44,61,47,64)(45,66,48,63)(49,56,52,59)(50,55,53,58)(51,60,54,57), (1,41,17,34)(2,42,18,35)(3,37,13,36)(4,38,14,31)(5,39,15,32)(6,40,16,33)(7,75,92,68)(8,76,93,69)(9,77,94,70)(10,78,95,71)(11,73,96,72)(12,74,91,67)(19,50,26,43)(20,51,27,44)(21,52,28,45)(22,53,29,46)(23,54,30,47)(24,49,25,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(20,24)(21,23)(25,27)(28,30)(32,36)(33,35)(37,39)(40,42)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(68,72)(69,71)(73,75)(76,78)(80,84)(81,83)(85,87)(88,90)>;
G:=Group( (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95), (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), (1,12,4,9)(2,11,5,8)(3,10,6,7)(13,95,16,92)(14,94,17,91)(15,93,18,96)(19,79,22,82)(20,84,23,81)(21,83,24,80)(25,87,28,90)(26,86,29,89)(27,85,30,88)(31,70,34,67)(32,69,35,72)(33,68,36,71)(37,78,40,75)(38,77,41,74)(39,76,42,73)(43,62,46,65)(44,61,47,64)(45,66,48,63)(49,56,52,59)(50,55,53,58)(51,60,54,57), (1,41,17,34)(2,42,18,35)(3,37,13,36)(4,38,14,31)(5,39,15,32)(6,40,16,33)(7,75,92,68)(8,76,93,69)(9,77,94,70)(10,78,95,71)(11,73,96,72)(12,74,91,67)(19,50,26,43)(20,51,27,44)(21,52,28,45)(22,53,29,46)(23,54,30,47)(24,49,25,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(20,24)(21,23)(25,27)(28,30)(32,36)(33,35)(37,39)(40,42)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(68,72)(69,71)(73,75)(76,78)(80,84)(81,83)(85,87)(88,90) );
G=PermutationGroup([(1,65),(2,66),(3,61),(4,62),(5,63),(6,64),(7,44),(8,45),(9,46),(10,47),(11,48),(12,43),(13,60),(14,55),(15,56),(16,57),(17,58),(18,59),(19,74),(20,75),(21,76),(22,77),(23,78),(24,73),(25,72),(26,67),(27,68),(28,69),(29,70),(30,71),(31,86),(32,87),(33,88),(34,89),(35,90),(36,85),(37,84),(38,79),(39,80),(40,81),(41,82),(42,83),(49,96),(50,91),(51,92),(52,93),(53,94),(54,95)], [(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)], [(1,12,4,9),(2,11,5,8),(3,10,6,7),(13,95,16,92),(14,94,17,91),(15,93,18,96),(19,79,22,82),(20,84,23,81),(21,83,24,80),(25,87,28,90),(26,86,29,89),(27,85,30,88),(31,70,34,67),(32,69,35,72),(33,68,36,71),(37,78,40,75),(38,77,41,74),(39,76,42,73),(43,62,46,65),(44,61,47,64),(45,66,48,63),(49,56,52,59),(50,55,53,58),(51,60,54,57)], [(1,41,17,34),(2,42,18,35),(3,37,13,36),(4,38,14,31),(5,39,15,32),(6,40,16,33),(7,75,92,68),(8,76,93,69),(9,77,94,70),(10,78,95,71),(11,73,96,72),(12,74,91,67),(19,50,26,43),(20,51,27,44),(21,52,28,45),(22,53,29,46),(23,54,30,47),(24,49,25,48),(55,86,62,79),(56,87,63,80),(57,88,64,81),(58,89,65,82),(59,90,66,83),(60,85,61,84)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,18),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(20,24),(21,23),(25,27),(28,30),(32,36),(33,35),(37,39),(40,42),(43,50),(44,49),(45,54),(46,53),(47,52),(48,51),(55,62),(56,61),(57,66),(58,65),(59,64),(60,63),(68,72),(69,71),(73,75),(76,78),(80,84),(81,83),(85,87),(88,90)])
Matrix representation ►G ⊆ 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
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],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,1,0],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0],[1,0,0,0,0,0,0,5,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 4U | 4V | 4W | 4X | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | Q8⋊3S3 |
| kernel | C2×Dic3⋊5D4 | Dic3⋊5D4 | C2×C4×Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | S3×C22×C4 | C22×D12 | C2×D12 | C2×C4⋊C4 | C2×Dic3 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 1 | 4 | 4 | 3 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times Dic_3\rtimes_5D_4
% in TeX
G:=Group("C2xDic3:5D4"); // GroupNames label
G:=SmallGroup(192,1062);
// by ID
G=gap.SmallGroup(192,1062);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations