direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C12×C22⋊C4, (C2×C42)⋊4C6, (C2×C6)⋊2C42, C2.1(D4×C12), (C22×C4)⋊8C12, C22⋊2(C4×C12), C6.103(C4×D4), (C23×C4).7C6, (C22×C12)⋊12C4, (C2×C12).533D4, (C23×C12).5C2, C24.28(C2×C6), C6.31(C2×C42), C22.27(C6×D4), C23.19(C2×C12), C2.C42⋊16C6, (C23×C6).82C22, C23.54(C22×C6), C6.51(C42⋊C2), C22.14(C22×C12), (C22×C6).441C23, (C22×C12).488C22, (C2×C4×C12)⋊2C2, C2.3(C2×C4×C12), (C2×C4)⋊6(C2×C12), (C2×C12)⋊28(C2×C4), C2.2(C6×C22⋊C4), (C2×C4).143(C3×D4), (C2×C6).594(C2×D4), C6.90(C2×C22⋊C4), (C2×C22⋊C4).14C6, (C6×C22⋊C4).34C2, (C22×C4).89(C2×C6), C2.2(C3×C42⋊C2), C22.13(C3×C4○D4), (C2×C6).203(C4○D4), (C2×C6).213(C22×C4), (C22×C6).109(C2×C4), (C3×C2.C42)⋊29C2, SmallGroup(192,810)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×C22⋊C4
G = < a,b,c,d | a12=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 370 in 258 conjugacy classes, 146 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×C12, C3×C22⋊C4, C22×C12, C22×C12, C22×C12, C23×C6, C4×C22⋊C4, C3×C2.C42, C2×C4×C12, C6×C22⋊C4, C23×C12, C12×C22⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C12, C3×C22⋊C4, C22×C12, C6×D4, C3×C4○D4, C4×C22⋊C4, C2×C4×C12, C6×C22⋊C4, C3×C42⋊C2, D4×C12, C12×C22⋊C4
►On 96 points
(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)
(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 85)(22 86)(23 87)(24 88)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 49)(12 50)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 85)(22 86)(23 87)(24 88)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 66 34 96)(2 67 35 85)(3 68 36 86)(4 69 25 87)(5 70 26 88)(6 71 27 89)(7 72 28 90)(8 61 29 91)(9 62 30 92)(10 63 31 93)(11 64 32 94)(12 65 33 95)(13 56 77 44)(14 57 78 45)(15 58 79 46)(16 59 80 47)(17 60 81 48)(18 49 82 37)(19 50 83 38)(20 51 84 39)(21 52 73 40)(22 53 74 41)(23 54 75 42)(24 55 76 43)
G:=sub<Sym(96)| (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), (13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,66,34,96)(2,67,35,85)(3,68,36,86)(4,69,25,87)(5,70,26,88)(6,71,27,89)(7,72,28,90)(8,61,29,91)(9,62,30,92)(10,63,31,93)(11,64,32,94)(12,65,33,95)(13,56,77,44)(14,57,78,45)(15,58,79,46)(16,59,80,47)(17,60,81,48)(18,49,82,37)(19,50,83,38)(20,51,84,39)(21,52,73,40)(22,53,74,41)(23,54,75,42)(24,55,76,43)>;
G:=Group( (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), (13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,85)(22,86)(23,87)(24,88)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,66,34,96)(2,67,35,85)(3,68,36,86)(4,69,25,87)(5,70,26,88)(6,71,27,89)(7,72,28,90)(8,61,29,91)(9,62,30,92)(10,63,31,93)(11,64,32,94)(12,65,33,95)(13,56,77,44)(14,57,78,45)(15,58,79,46)(16,59,80,47)(17,60,81,48)(18,49,82,37)(19,50,83,38)(20,51,84,39)(21,52,73,40)(22,53,74,41)(23,54,75,42)(24,55,76,43) );
G=PermutationGroup([[(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)], [(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,85),(22,86),(23,87),(24,88),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,49),(12,50),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,85),(22,86),(23,87),(24,88),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(1,66,34,96),(2,67,35,85),(3,68,36,86),(4,69,25,87),(5,70,26,88),(6,71,27,89),(7,72,28,90),(8,61,29,91),(9,62,30,92),(10,63,31,93),(11,64,32,94),(12,65,33,95),(13,56,77,44),(14,57,78,45),(15,58,79,46),(16,59,80,47),(17,60,81,48),(18,49,82,37),(19,50,83,38),(20,51,84,39),(21,52,73,40),(22,53,74,41),(23,54,75,42),(24,55,76,43)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4AB | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C12×C22⋊C4 | C3×C2.C42 | C2×C4×C12 | C6×C22⋊C4 | C23×C12 | C4×C22⋊C4 | C3×C22⋊C4 | C22×C12 | C2.C42 | C2×C42 | C2×C22⋊C4 | C23×C4 | C22⋊C4 | C22×C4 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 16 | 8 | 4 | 4 | 4 | 2 | 32 | 16 | 4 | 4 | 8 | 8 |
Matrix representation of C12×C22⋊C4 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 5 |
| 0 | 0 | 3 | 5 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[12,0,0,0,0,12,0,0,0,0,1,2,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,5,0,0,0,0,8,3,0,0,5,5] >;
C12×C22⋊C4 in GAP, Magma, Sage, TeX
C_{12}\times C_2^2\rtimes C_4 % in TeX
G:=Group("C12xC2^2:C4"); // GroupNames label
G:=SmallGroup(192,810);
// by ID
G=gap.SmallGroup(192,810);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,268]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations