direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.48D4, C23⋊5Dic6, C24.78D6, (C22×C6)⋊7Q8, C6⋊4(C22⋊Q8), C12.424(C2×D4), (C2×C12).477D4, (C23×C4).16S3, C22⋊4(C2×Dic6), C6.19(C22×Q8), (C23×C12).12C2, (C2×C6).282C24, C4⋊Dic3⋊63C22, (C22×C4).462D6, C6.130(C22×D4), (C2×C12).703C23, Dic3⋊C4⋊43C22, (C2×Dic6)⋊58C22, (C22×Dic6)⋊12C2, C2.20(C22×Dic6), C22.79(C4○D12), (C23×C6).104C22, C22.301(S3×C23), C23.241(C22×S3), (C22×C6).411C23, (C22×C12).528C22, (C2×Dic3).148C23, C6.D4.129C22, (C22×Dic3).160C22, (C2×C6)⋊6(C2×Q8), C3⋊5(C2×C22⋊Q8), C6.59(C2×C4○D4), (C2×C4⋊Dic3)⋊28C2, C2.69(C2×C4○D12), (C2×C6).571(C2×D4), C4.120(C2×C3⋊D4), C2.5(C22×C3⋊D4), (C2×Dic3⋊C4)⋊17C2, (C2×C6).110(C4○D4), (C2×C4).262(C3⋊D4), (C2×C4).656(C22×S3), C22.100(C2×C3⋊D4), (C2×C6.D4).23C2, SmallGroup(192,1343)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 664 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×26], Q8 [×8], C23, C23 [×6], C23 [×4], Dic3 [×8], C12 [×4], C12 [×2], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×8], C24, Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×8], C2×C12 [×10], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], C23×C6, C2×C22⋊Q8, C2×Dic3⋊C4 [×2], C12.48D4 [×8], C2×C4⋊Dic3, C2×C6.D4 [×2], C22×Dic6, C23×C12, C2×C12.48D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C3⋊D4 [×4], C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×Dic6 [×6], C4○D12 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C22⋊Q8, C12.48D4 [×4], C22×Dic6, C2×C4○D12, C22×C3⋊D4, C2×C12.48D4
Generators and relations
G = < a,b,c,d | a2=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-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 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 49)(47 50)(48 51)
(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 35 53 71)(2 34 54 70)(3 33 55 69)(4 32 56 68)(5 31 57 67)(6 30 58 66)(7 29 59 65)(8 28 60 64)(9 27 49 63)(10 26 50 62)(11 25 51 61)(12 36 52 72)(13 73 94 45)(14 84 95 44)(15 83 96 43)(16 82 85 42)(17 81 86 41)(18 80 87 40)(19 79 88 39)(20 78 89 38)(21 77 90 37)(22 76 91 48)(23 75 92 47)(24 74 93 46)
(1 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 72 12 66)(13 73 19 79)(14 84 20 78)(15 83 21 77)(16 82 22 76)(17 81 23 75)(18 80 24 74)(25 51 31 57)(26 50 32 56)(27 49 33 55)(28 60 34 54)(29 59 35 53)(30 58 36 52)(37 96 43 90)(38 95 44 89)(39 94 45 88)(40 93 46 87)(41 92 47 86)(42 91 48 85)
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)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,49)(47,50)(48,51), (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), (1,35,53,71)(2,34,54,70)(3,33,55,69)(4,32,56,68)(5,31,57,67)(6,30,58,66)(7,29,59,65)(8,28,60,64)(9,27,49,63)(10,26,50,62)(11,25,51,61)(12,36,52,72)(13,73,94,45)(14,84,95,44)(15,83,96,43)(16,82,85,42)(17,81,86,41)(18,80,87,40)(19,79,88,39)(20,78,89,38)(21,77,90,37)(22,76,91,48)(23,75,92,47)(24,74,93,46), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,73,19,79)(14,84,20,78)(15,83,21,77)(16,82,22,76)(17,81,23,75)(18,80,24,74)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52)(37,96,43,90)(38,95,44,89)(39,94,45,88)(40,93,46,87)(41,92,47,86)(42,91,48,85)>;
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)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,49)(47,50)(48,51), (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), (1,35,53,71)(2,34,54,70)(3,33,55,69)(4,32,56,68)(5,31,57,67)(6,30,58,66)(7,29,59,65)(8,28,60,64)(9,27,49,63)(10,26,50,62)(11,25,51,61)(12,36,52,72)(13,73,94,45)(14,84,95,44)(15,83,96,43)(16,82,85,42)(17,81,86,41)(18,80,87,40)(19,79,88,39)(20,78,89,38)(21,77,90,37)(22,76,91,48)(23,75,92,47)(24,74,93,46), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,73,19,79)(14,84,20,78)(15,83,21,77)(16,82,22,76)(17,81,23,75)(18,80,24,74)(25,51,31,57)(26,50,32,56)(27,49,33,55)(28,60,34,54)(29,59,35,53)(30,58,36,52)(37,96,43,90)(38,95,44,89)(39,94,45,88)(40,93,46,87)(41,92,47,86)(42,91,48,85) );
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),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,61),(23,62),(24,63),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,49),(47,50),(48,51)], [(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)], [(1,35,53,71),(2,34,54,70),(3,33,55,69),(4,32,56,68),(5,31,57,67),(6,30,58,66),(7,29,59,65),(8,28,60,64),(9,27,49,63),(10,26,50,62),(11,25,51,61),(12,36,52,72),(13,73,94,45),(14,84,95,44),(15,83,96,43),(16,82,85,42),(17,81,86,41),(18,80,87,40),(19,79,88,39),(20,78,89,38),(21,77,90,37),(22,76,91,48),(23,75,92,47),(24,74,93,46)], [(1,65,7,71),(2,64,8,70),(3,63,9,69),(4,62,10,68),(5,61,11,67),(6,72,12,66),(13,73,19,79),(14,84,20,78),(15,83,21,77),(16,82,22,76),(17,81,23,75),(18,80,24,74),(25,51,31,57),(26,50,32,56),(27,49,33,55),(28,60,34,54),(29,59,35,53),(30,58,36,52),(37,96,43,90),(38,95,44,89),(39,94,45,88),(40,93,46,87),(41,92,47,86),(42,91,48,85)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 |
| kernel | C2×C12.48D4 | C2×Dic3⋊C4 | C12.48D4 | C2×C4⋊Dic3 | C2×C6.D4 | C22×Dic6 | C23×C12 | C23×C4 | C2×C12 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 4 | 4 | 6 | 1 | 4 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}._{48}D_4 % in TeX
G:=Group("C2xC12.48D4"); // GroupNames label
G:=SmallGroup(192,1343);
// by ID
G=gap.SmallGroup(192,1343);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,675,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations