direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12⋊D4, C4⋊C4⋊38D6, D6⋊2(C2×D4), (C2×C12)⋊8D4, C4⋊3(C2×D12), C12⋊4(C2×D4), (C2×C4)⋊10D12, C6⋊2(C4⋊D4), C6.9(C22×D4), D6⋊C4⋊50C22, (C22×S3)⋊10D4, (C22×D12)⋊7C2, (C2×C6).50C24, (C2×D12)⋊45C22, C22.133(S3×D4), C22.67(C2×D12), (C22×C4).376D6, C2.11(C22×D12), (C2×C12).487C23, C22.84(S3×C23), (C22×C6).399C23, C23.338(C22×S3), (C22×S3).156C23, (S3×C23).100C22, (C22×C12).217C22, C22.36(Q8⋊3S3), (C2×Dic3).188C23, (C22×Dic3).213C22, C3⋊2(C2×C4⋊D4), (C6×C4⋊C4)⋊12C2, (C2×C4⋊C4)⋊15S3, C2.15(C2×S3×D4), (S3×C22×C4)⋊1C2, (C2×D6⋊C4)⋊20C2, (S3×C2×C4)⋊55C22, (C3×C4⋊C4)⋊46C22, C6.109(C2×C4○D4), (C2×C6).174(C2×D4), C2.7(C2×Q8⋊3S3), (C2×C6).197(C4○D4), (C2×C4).141(C22×S3), SmallGroup(192,1065)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1304 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×6], C22 [×36], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×16], D4 [×24], C23, C23 [×26], Dic3 [×2], C12 [×4], C12 [×4], D6 [×4], D6 [×32], C2×C6, C2×C6 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4 [×24], C24 [×3], C4×S3 [×8], D12 [×24], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×10], C2×C12 [×4], C22×S3 [×10], C22×S3 [×16], C22×C6, C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], D6⋊C4 [×8], C3×C4⋊C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C2×D12 [×12], C2×D12 [×12], C22×Dic3, C22×C12, C22×C12 [×2], S3×C23, S3×C23 [×2], C2×C4⋊D4, C12⋊D4 [×8], C2×D6⋊C4 [×2], C6×C4⋊C4, S3×C22×C4, C22×D12, C22×D12 [×2], C2×C12⋊D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×D12 [×6], S3×D4 [×2], Q8⋊3S3 [×2], S3×C23, C2×C4⋊D4, C12⋊D4 [×4], C22×D12, C2×S3×D4, C2×Q8⋊3S3, C2×C12⋊D4
Generators and relations
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >
►On 96 points
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 73)(9 74)(10 75)(11 76)(12 77)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 60)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)
(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 53 43 18)(2 60 44 13)(3 55 45 20)(4 50 46 15)(5 57 47 22)(6 52 48 17)(7 59 37 24)(8 54 38 19)(9 49 39 14)(10 56 40 21)(11 51 41 16)(12 58 42 23)(25 86 67 79)(26 93 68 74)(27 88 69 81)(28 95 70 76)(29 90 71 83)(30 85 72 78)(31 92 61 73)(32 87 62 80)(33 94 63 75)(34 89 64 82)(35 96 65 77)(36 91 66 84)
(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 96)(9 95)(10 94)(11 93)(12 92)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 72)(25 52)(26 51)(27 50)(28 49)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 78)(38 77)(39 76)(40 75)(41 74)(42 73)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)
G:=sub<Sym(96)| (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,73)(9,74)(10,75)(11,76)(12,77)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90), (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), (1,53,43,18)(2,60,44,13)(3,55,45,20)(4,50,46,15)(5,57,47,22)(6,52,48,17)(7,59,37,24)(8,54,38,19)(9,49,39,14)(10,56,40,21)(11,51,41,16)(12,58,42,23)(25,86,67,79)(26,93,68,74)(27,88,69,81)(28,95,70,76)(29,90,71,83)(30,85,72,78)(31,92,61,73)(32,87,62,80)(33,94,63,75)(34,89,64,82)(35,96,65,77)(36,91,66,84), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,72)(25,52)(26,51)(27,50)(28,49)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)>;
G:=Group( (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,73)(9,74)(10,75)(11,76)(12,77)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90), (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), (1,53,43,18)(2,60,44,13)(3,55,45,20)(4,50,46,15)(5,57,47,22)(6,52,48,17)(7,59,37,24)(8,54,38,19)(9,49,39,14)(10,56,40,21)(11,51,41,16)(12,58,42,23)(25,86,67,79)(26,93,68,74)(27,88,69,81)(28,95,70,76)(29,90,71,83)(30,85,72,78)(31,92,61,73)(32,87,62,80)(33,94,63,75)(34,89,64,82)(35,96,65,77)(36,91,66,84), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,72)(25,52)(26,51)(27,50)(28,49)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79) );
G=PermutationGroup([(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,73),(9,74),(10,75),(11,76),(12,77),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,60),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90)], [(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)], [(1,53,43,18),(2,60,44,13),(3,55,45,20),(4,50,46,15),(5,57,47,22),(6,52,48,17),(7,59,37,24),(8,54,38,19),(9,49,39,14),(10,56,40,21),(11,51,41,16),(12,58,42,23),(25,86,67,79),(26,93,68,74),(27,88,69,81),(28,95,70,76),(29,90,71,83),(30,85,72,78),(31,92,61,73),(32,87,62,80),(33,94,63,75),(34,89,64,82),(35,96,65,77),(36,91,66,84)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,96),(9,95),(10,94),(11,93),(12,92),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,72),(25,52),(26,51),(27,50),(28,49),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,78),(38,77),(39,76),(40,75),(41,74),(42,73),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,12,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,3,0,0,0,0,8,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,3,0,0,0,0,0,1] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D12 | S3×D4 | Q8⋊3S3 |
| kernel | C2×C12⋊D4 | C12⋊D4 | C2×D6⋊C4 | C6×C4⋊C4 | S3×C22×C4 | C22×D12 | C2×C4⋊C4 | C2×C12 | C22×S3 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 1 | 3 | 1 | 4 | 4 | 4 | 3 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}\rtimes D_4 % in TeX
G:=Group("C2xC12:D4"); // GroupNames label
G:=SmallGroup(192,1065);
// by ID
G=gap.SmallGroup(192,1065);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations