metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.229D6, (C4×D4)⋊19S3, (D4×C12)⋊21C2, (S3×C42)⋊5C2, C4⋊C4.285D6, C4.D12⋊48C2, (C4×Dic6)⋊33C2, D6.2(C4○D4), (C2×D4).218D6, C23.9D6⋊54C2, D6⋊3D4.14C2, C4.44(C4○D12), (C2×C6).101C24, D6⋊C4.99C22, C22⋊C4.114D6, C12.3Q8⋊46C2, (C22×C4).228D6, C12.310(C4○D4), C23.12D6⋊32C2, (C2×C12).161C23, (C4×C12).156C22, (C6×D4).261C22, C23.26D6⋊8C2, C23.8D6⋊50C2, C4.137(D4⋊2S3), C4⋊Dic3.300C22, (C22×C6).171C23, C23.108(C22×S3), C22.126(S3×C23), Dic3⋊C4.112C22, (C22×S3).174C23, (C22×C12).110C22, C3⋊4(C23.36C23), (C2×Dic3).208C23, (C2×Dic6).288C22, (C4×Dic3).293C22, C6.D4.106C22, (C4×C3⋊D4)⋊5C2, C2.24(S3×C4○D4), C6.141(C2×C4○D4), C2.50(C2×C4○D12), C2.23(C2×D4⋊2S3), (S3×C2×C4).293C22, (C3×C4⋊C4).330C22, (C2×C4).161(C22×S3), (C2×C3⋊D4).115C22, (C3×C22⋊C4).125C22, SmallGroup(192,1116)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 520 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, Dic3 [×7], C12 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic6 [×2], C4×S3 [×6], C2×Dic3 [×3], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×S3, C22×C6 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×3], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×2], C6.D4 [×6], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×3], C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, C23.36C23, C4×Dic6, S3×C42, C23.8D6 [×2], C23.9D6 [×2], C12.3Q8, C4.D12, C23.26D6 [×2], C4×C3⋊D4 [×2], C23.12D6, D6⋊3D4, D4×C12, C42.229D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], C4○D12 [×2], D4⋊2S3 [×2], S3×C23, C23.36C23, C2×C4○D12, C2×D4⋊2S3, S3×C4○D4, C42.229D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c-1 >
►On 96 points
(1 64 19 80)(2 68 20 54)(3 66 21 82)(4 70 22 50)(5 62 23 84)(6 72 24 52)(7 51 27 71)(8 79 28 63)(9 53 29 67)(10 81 30 65)(11 49 25 69)(12 83 26 61)(13 35 86 47)(14 55 87 41)(15 31 88 43)(16 57 89 37)(17 33 90 45)(18 59 85 39)(32 92 44 78)(34 94 46 74)(36 96 48 76)(38 73 58 93)(40 75 60 95)(42 77 56 91)
(1 42 9 31)(2 37 10 32)(3 38 11 33)(4 39 12 34)(5 40 7 35)(6 41 8 36)(13 84 95 71)(14 79 96 72)(15 80 91 67)(16 81 92 68)(17 82 93 69)(18 83 94 70)(19 56 29 43)(20 57 30 44)(21 58 25 45)(22 59 26 46)(23 60 27 47)(24 55 28 48)(49 90 66 73)(50 85 61 74)(51 86 62 75)(52 87 63 76)(53 88 64 77)(54 89 65 78)
(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 6 19 24)(2 23 20 5)(3 4 21 22)(7 10 27 30)(8 29 28 9)(11 12 25 26)(13 16 86 89)(14 88 87 15)(17 18 90 85)(31 36 43 48)(32 47 44 35)(33 34 45 46)(37 60 57 40)(38 39 58 59)(41 56 55 42)(49 50 69 70)(51 54 71 68)(52 67 72 53)(61 82 83 66)(62 65 84 81)(63 80 79 64)(73 74 93 94)(75 78 95 92)(76 91 96 77)
G:=sub<Sym(96)| (1,64,19,80)(2,68,20,54)(3,66,21,82)(4,70,22,50)(5,62,23,84)(6,72,24,52)(7,51,27,71)(8,79,28,63)(9,53,29,67)(10,81,30,65)(11,49,25,69)(12,83,26,61)(13,35,86,47)(14,55,87,41)(15,31,88,43)(16,57,89,37)(17,33,90,45)(18,59,85,39)(32,92,44,78)(34,94,46,74)(36,96,48,76)(38,73,58,93)(40,75,60,95)(42,77,56,91), (1,42,9,31)(2,37,10,32)(3,38,11,33)(4,39,12,34)(5,40,7,35)(6,41,8,36)(13,84,95,71)(14,79,96,72)(15,80,91,67)(16,81,92,68)(17,82,93,69)(18,83,94,70)(19,56,29,43)(20,57,30,44)(21,58,25,45)(22,59,26,46)(23,60,27,47)(24,55,28,48)(49,90,66,73)(50,85,61,74)(51,86,62,75)(52,87,63,76)(53,88,64,77)(54,89,65,78), (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), (1,6,19,24)(2,23,20,5)(3,4,21,22)(7,10,27,30)(8,29,28,9)(11,12,25,26)(13,16,86,89)(14,88,87,15)(17,18,90,85)(31,36,43,48)(32,47,44,35)(33,34,45,46)(37,60,57,40)(38,39,58,59)(41,56,55,42)(49,50,69,70)(51,54,71,68)(52,67,72,53)(61,82,83,66)(62,65,84,81)(63,80,79,64)(73,74,93,94)(75,78,95,92)(76,91,96,77)>;
G:=Group( (1,64,19,80)(2,68,20,54)(3,66,21,82)(4,70,22,50)(5,62,23,84)(6,72,24,52)(7,51,27,71)(8,79,28,63)(9,53,29,67)(10,81,30,65)(11,49,25,69)(12,83,26,61)(13,35,86,47)(14,55,87,41)(15,31,88,43)(16,57,89,37)(17,33,90,45)(18,59,85,39)(32,92,44,78)(34,94,46,74)(36,96,48,76)(38,73,58,93)(40,75,60,95)(42,77,56,91), (1,42,9,31)(2,37,10,32)(3,38,11,33)(4,39,12,34)(5,40,7,35)(6,41,8,36)(13,84,95,71)(14,79,96,72)(15,80,91,67)(16,81,92,68)(17,82,93,69)(18,83,94,70)(19,56,29,43)(20,57,30,44)(21,58,25,45)(22,59,26,46)(23,60,27,47)(24,55,28,48)(49,90,66,73)(50,85,61,74)(51,86,62,75)(52,87,63,76)(53,88,64,77)(54,89,65,78), (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), (1,6,19,24)(2,23,20,5)(3,4,21,22)(7,10,27,30)(8,29,28,9)(11,12,25,26)(13,16,86,89)(14,88,87,15)(17,18,90,85)(31,36,43,48)(32,47,44,35)(33,34,45,46)(37,60,57,40)(38,39,58,59)(41,56,55,42)(49,50,69,70)(51,54,71,68)(52,67,72,53)(61,82,83,66)(62,65,84,81)(63,80,79,64)(73,74,93,94)(75,78,95,92)(76,91,96,77) );
G=PermutationGroup([(1,64,19,80),(2,68,20,54),(3,66,21,82),(4,70,22,50),(5,62,23,84),(6,72,24,52),(7,51,27,71),(8,79,28,63),(9,53,29,67),(10,81,30,65),(11,49,25,69),(12,83,26,61),(13,35,86,47),(14,55,87,41),(15,31,88,43),(16,57,89,37),(17,33,90,45),(18,59,85,39),(32,92,44,78),(34,94,46,74),(36,96,48,76),(38,73,58,93),(40,75,60,95),(42,77,56,91)], [(1,42,9,31),(2,37,10,32),(3,38,11,33),(4,39,12,34),(5,40,7,35),(6,41,8,36),(13,84,95,71),(14,79,96,72),(15,80,91,67),(16,81,92,68),(17,82,93,69),(18,83,94,70),(19,56,29,43),(20,57,30,44),(21,58,25,45),(22,59,26,46),(23,60,27,47),(24,55,28,48),(49,90,66,73),(50,85,61,74),(51,86,62,75),(52,87,63,76),(53,88,64,77),(54,89,65,78)], [(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)], [(1,6,19,24),(2,23,20,5),(3,4,21,22),(7,10,27,30),(8,29,28,9),(11,12,25,26),(13,16,86,89),(14,88,87,15),(17,18,90,85),(31,36,43,48),(32,47,44,35),(33,34,45,46),(37,60,57,40),(38,39,58,59),(41,56,55,42),(49,50,69,70),(51,54,71,68),(52,67,72,53),(61,82,83,66),(62,65,84,81),(63,80,79,64),(73,74,93,94),(75,78,95,92),(76,91,96,77)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,9,1,0,0,0,0,11,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,6,10,0,0,0,0,3,7] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | D4⋊2S3 | S3×C4○D4 |
| kernel | C42.229D6 | C4×Dic6 | S3×C42 | C23.8D6 | C23.9D6 | C12.3Q8 | C4.D12 | C23.26D6 | C4×C3⋊D4 | C23.12D6 | D6⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{229}D_6 % in TeX
G:=Group("C4^2.229D6"); // GroupNames label
G:=SmallGroup(192,1116);
// by ID
G=gap.SmallGroup(192,1116);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,794,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations