direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×D12, C42⋊40D6, C6⋊1(C4×D4), C12⋊11(C2×D4), (C2×C12)⋊32D4, (C2×C42)⋊8S3, C12⋊5(C22×C4), D6⋊1(C22×C4), C6.4(C23×C4), C6.2(C22×D4), (C4×C12)⋊53C22, D6⋊C4⋊75C22, (C2×C6).17C24, C2.1(C22×D12), C4⋊Dic3⋊81C22, (C22×C4).484D6, C22.63(C2×D12), (C2×C12).875C23, (C22×D12).20C2, C22.14(S3×C23), (C2×D12).284C22, C22.68(C4○D12), (S3×C23).92C22, C23.324(C22×S3), (C22×C6).379C23, (C22×S3).146C23, (C22×C12).503C22, (C2×Dic3).173C23, (C22×Dic3).201C22, C3⋊1(C2×C4×D4), C4⋊3(S3×C2×C4), (C2×C4×C12)⋊11C2, (C2×C4)⋊12(C4×S3), C6.5(C2×C4○D4), (C2×C12)⋊29(C2×C4), (C2×D6⋊C4)⋊45C2, C2.6(S3×C22×C4), (S3×C2×C4)⋊62C22, (S3×C22×C4)⋊14C2, C2.3(C2×C4○D12), C22.69(S3×C2×C4), (C22×S3)⋊9(C2×C4), (C2×C4⋊Dic3)⋊49C2, (C2×C6).169(C2×D4), (C2×C6).96(C4○D4), (C2×C4).817(C22×S3), (C2×C6).147(C22×C4), SmallGroup(192,1032)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1048 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×8], C4 [×6], C22, C22 [×6], C22 [×32], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×14], C2×C4 [×26], D4 [×16], C23, C23 [×20], Dic3 [×4], C12 [×8], C12 [×2], D6 [×8], D6 [×24], C2×C6, C2×C6 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C24 [×2], C4×S3 [×16], D12 [×16], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×14], C2×C12 [×2], 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], C4×C12 [×4], S3×C2×C4 [×8], S3×C2×C4 [×8], C2×D12 [×12], C22×Dic3 [×2], C22×C12 [×3], S3×C23 [×2], C2×C4×D4, C4×D12 [×8], C2×C4⋊Dic3, C2×D6⋊C4 [×2], C2×C4×C12, S3×C22×C4 [×2], C22×D12, C2×C4×D12
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], D12 [×4], C22×S3 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, S3×C2×C4 [×6], C2×D12 [×6], C4○D12 [×2], S3×C23, C2×C4×D4, C4×D12 [×4], S3×C22×C4, C22×D12, C2×C4○D12, C2×C4×D12
Generators and relations
G = < a,b,c,d | a2=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 96 points
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 61)(58 62)(59 63)(60 64)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(1 57 46 76)(2 58 47 77)(3 59 48 78)(4 60 37 79)(5 49 38 80)(6 50 39 81)(7 51 40 82)(8 52 41 83)(9 53 42 84)(10 54 43 73)(11 55 44 74)(12 56 45 75)(13 92 34 71)(14 93 35 72)(15 94 36 61)(16 95 25 62)(17 96 26 63)(18 85 27 64)(19 86 28 65)(20 87 29 66)(21 88 30 67)(22 89 31 68)(23 90 32 69)(24 91 33 70)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 41)(14 40)(15 39)(16 38)(17 37)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(49 62)(50 61)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)
G:=sub<Sym(96)| (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,44)(14,45)(15,46)(16,47)(17,48)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,61)(58,62)(59,63)(60,64)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,57,46,76)(2,58,47,77)(3,59,48,78)(4,60,37,79)(5,49,38,80)(6,50,39,81)(7,51,40,82)(8,52,41,83)(9,53,42,84)(10,54,43,73)(11,55,44,74)(12,56,45,75)(13,92,34,71)(14,93,35,72)(15,94,36,61)(16,95,25,62)(17,96,26,63)(18,85,27,64)(19,86,28,65)(20,87,29,66)(21,88,30,67)(22,89,31,68)(23,90,32,69)(24,91,33,70), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)>;
G:=Group( (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,44)(14,45)(15,46)(16,47)(17,48)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,61)(58,62)(59,63)(60,64)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,57,46,76)(2,58,47,77)(3,59,48,78)(4,60,37,79)(5,49,38,80)(6,50,39,81)(7,51,40,82)(8,52,41,83)(9,53,42,84)(10,54,43,73)(11,55,44,74)(12,56,45,75)(13,92,34,71)(14,93,35,72)(15,94,36,61)(16,95,25,62)(17,96,26,63)(18,85,27,64)(19,86,28,65)(20,87,29,66)(21,88,30,67)(22,89,31,68)(23,90,32,69)(24,91,33,70), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91) );
G=PermutationGroup([(1,36),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,44),(14,45),(15,46),(16,47),(17,48),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,61),(58,62),(59,63),(60,64),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(1,57,46,76),(2,58,47,77),(3,59,48,78),(4,60,37,79),(5,49,38,80),(6,50,39,81),(7,51,40,82),(8,52,41,83),(9,53,42,84),(10,54,43,73),(11,55,44,74),(12,56,45,75),(13,92,34,71),(14,93,35,72),(15,94,36,61),(16,95,25,62),(17,96,26,63),(18,85,27,64),(19,86,28,65),(20,87,29,66),(21,88,30,67),(22,89,31,68),(23,90,32,69),(24,91,33,70)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,41),(14,40),(15,39),(16,38),(17,37),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(49,62),(50,61),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 7 | 10 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,3,3,0,0,10,6],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 |
| kernel | C2×C4×D12 | C4×D12 | C2×C4⋊Dic3 | C2×D6⋊C4 | C2×C4×C12 | S3×C22×C4 | C22×D12 | C2×D12 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 1 | 4 | 4 | 3 | 4 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times D_{12} % in TeX
G:=Group("C2xC4xD12"); // GroupNames label
G:=SmallGroup(192,1032);
// by ID
G=gap.SmallGroup(192,1032);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations