metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.197D6, (D4×Dic3)⋊27C2, (C2×D4).159D6, C22⋊C4.66D6, Dic3.Q8⋊24C2, (C2×C6).193C24, (C2×C12).67C23, D6⋊C4.31C22, (C22×C4).337D6, Dic3⋊4D4⋊17C2, Dic6⋊C4⋊30C2, Dic3.8(C4○D4), C23.14D6.1C2, (C6×D4).131C22, C23.8D6⋊27C2, C22.D4⋊17S3, (C22×C6).29C23, C23.34(C22×S3), C23.11D6⋊30C2, Dic3.D4⋊28C2, C23.16D6⋊11C2, C23.28D6⋊19C2, Dic3⋊C4.38C22, (C22×S3).84C23, C4⋊Dic3.224C22, C22.214(S3×C23), (C2×Dic3).98C23, C22.11(D4⋊2S3), (C22×C12).367C22, C3⋊8(C23.36C23), (C4×Dic3).120C22, (C2×Dic6).166C22, C6.D4.39C22, (C22×Dic3).226C22, C4⋊C4⋊7S3⋊31C2, C4⋊C4⋊S3⋊26C2, (C2×C4×Dic3)⋊36C2, C2.57(S3×C4○D4), C6.169(C2×C4○D4), (C2×C6).45(C4○D4), C2.51(C2×D4⋊2S3), (S3×C2×C4).109C22, (C2×C4).58(C22×S3), (C3×C4⋊C4).173C22, (C3×C22.D4)⋊3C2, (C2×C3⋊D4).45C22, (C3×C22⋊C4).48C22, SmallGroup(192,1208)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 528 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×14], C22, C22 [×2], C22 [×8], S3, C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, Dic3 [×4], Dic3 [×5], C12 [×5], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×7], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3, C22×C6 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×6], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×3], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4, C22×Dic3 [×3], C2×C3⋊D4 [×2], C22×C12, C6×D4, C23.36C23, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4 [×2], C23.11D6, Dic6⋊C4, Dic3.Q8, C4⋊C4⋊7S3, C4⋊C4⋊S3, C2×C4×Dic3, C23.28D6, D4×Dic3, C23.14D6, C3×C22.D4, C4⋊C4.197D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], D4⋊2S3 [×2], S3×C23, C23.36C23, C2×D4⋊2S3, S3×C4○D4 [×2], C4⋊C4.197D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, bab-1=a-1, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
►On 96 points
(1 10 73 54)(2 27 74 95)(3 12 75 50)(4 29 76 91)(5 8 77 52)(6 25 78 93)(7 32 51 58)(9 34 53 60)(11 36 49 56)(13 72 81 46)(14 23 82 85)(15 68 83 48)(16 19 84 87)(17 70 79 44)(18 21 80 89)(20 41 88 61)(22 37 90 63)(24 39 86 65)(26 35 94 55)(28 31 96 57)(30 33 92 59)(38 47 64 67)(40 43 66 69)(42 45 62 71)
(1 67 55 23)(2 68 56 24)(3 69 57 19)(4 70 58 20)(5 71 59 21)(6 72 60 22)(7 61 29 17)(8 62 30 18)(9 63 25 13)(10 64 26 14)(11 65 27 15)(12 66 28 16)(31 87 75 43)(32 88 76 44)(33 89 77 45)(34 90 78 46)(35 85 73 47)(36 86 74 48)(37 93 81 53)(38 94 82 54)(39 95 83 49)(40 96 84 50)(41 91 79 51)(42 92 80 52)
(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 25 73 93)(2 30 74 92)(3 29 75 91)(4 28 76 96)(5 27 77 95)(6 26 78 94)(7 31 51 57)(8 36 52 56)(9 35 53 55)(10 34 54 60)(11 33 49 59)(12 32 50 58)(13 47 81 67)(14 46 82 72)(15 45 83 71)(16 44 84 70)(17 43 79 69)(18 48 80 68)(19 61 87 41)(20 66 88 40)(21 65 89 39)(22 64 90 38)(23 63 85 37)(24 62 86 42)
G:=sub<Sym(96)| (1,10,73,54)(2,27,74,95)(3,12,75,50)(4,29,76,91)(5,8,77,52)(6,25,78,93)(7,32,51,58)(9,34,53,60)(11,36,49,56)(13,72,81,46)(14,23,82,85)(15,68,83,48)(16,19,84,87)(17,70,79,44)(18,21,80,89)(20,41,88,61)(22,37,90,63)(24,39,86,65)(26,35,94,55)(28,31,96,57)(30,33,92,59)(38,47,64,67)(40,43,66,69)(42,45,62,71), (1,67,55,23)(2,68,56,24)(3,69,57,19)(4,70,58,20)(5,71,59,21)(6,72,60,22)(7,61,29,17)(8,62,30,18)(9,63,25,13)(10,64,26,14)(11,65,27,15)(12,66,28,16)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,25,73,93)(2,30,74,92)(3,29,75,91)(4,28,76,96)(5,27,77,95)(6,26,78,94)(7,31,51,57)(8,36,52,56)(9,35,53,55)(10,34,54,60)(11,33,49,59)(12,32,50,58)(13,47,81,67)(14,46,82,72)(15,45,83,71)(16,44,84,70)(17,43,79,69)(18,48,80,68)(19,61,87,41)(20,66,88,40)(21,65,89,39)(22,64,90,38)(23,63,85,37)(24,62,86,42)>;
G:=Group( (1,10,73,54)(2,27,74,95)(3,12,75,50)(4,29,76,91)(5,8,77,52)(6,25,78,93)(7,32,51,58)(9,34,53,60)(11,36,49,56)(13,72,81,46)(14,23,82,85)(15,68,83,48)(16,19,84,87)(17,70,79,44)(18,21,80,89)(20,41,88,61)(22,37,90,63)(24,39,86,65)(26,35,94,55)(28,31,96,57)(30,33,92,59)(38,47,64,67)(40,43,66,69)(42,45,62,71), (1,67,55,23)(2,68,56,24)(3,69,57,19)(4,70,58,20)(5,71,59,21)(6,72,60,22)(7,61,29,17)(8,62,30,18)(9,63,25,13)(10,64,26,14)(11,65,27,15)(12,66,28,16)(31,87,75,43)(32,88,76,44)(33,89,77,45)(34,90,78,46)(35,85,73,47)(36,86,74,48)(37,93,81,53)(38,94,82,54)(39,95,83,49)(40,96,84,50)(41,91,79,51)(42,92,80,52), (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), (1,25,73,93)(2,30,74,92)(3,29,75,91)(4,28,76,96)(5,27,77,95)(6,26,78,94)(7,31,51,57)(8,36,52,56)(9,35,53,55)(10,34,54,60)(11,33,49,59)(12,32,50,58)(13,47,81,67)(14,46,82,72)(15,45,83,71)(16,44,84,70)(17,43,79,69)(18,48,80,68)(19,61,87,41)(20,66,88,40)(21,65,89,39)(22,64,90,38)(23,63,85,37)(24,62,86,42) );
G=PermutationGroup([(1,10,73,54),(2,27,74,95),(3,12,75,50),(4,29,76,91),(5,8,77,52),(6,25,78,93),(7,32,51,58),(9,34,53,60),(11,36,49,56),(13,72,81,46),(14,23,82,85),(15,68,83,48),(16,19,84,87),(17,70,79,44),(18,21,80,89),(20,41,88,61),(22,37,90,63),(24,39,86,65),(26,35,94,55),(28,31,96,57),(30,33,92,59),(38,47,64,67),(40,43,66,69),(42,45,62,71)], [(1,67,55,23),(2,68,56,24),(3,69,57,19),(4,70,58,20),(5,71,59,21),(6,72,60,22),(7,61,29,17),(8,62,30,18),(9,63,25,13),(10,64,26,14),(11,65,27,15),(12,66,28,16),(31,87,75,43),(32,88,76,44),(33,89,77,45),(34,90,78,46),(35,85,73,47),(36,86,74,48),(37,93,81,53),(38,94,82,54),(39,95,83,49),(40,96,84,50),(41,91,79,51),(42,92,80,52)], [(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)], [(1,25,73,93),(2,30,74,92),(3,29,75,91),(4,28,76,96),(5,27,77,95),(6,26,78,94),(7,31,51,57),(8,36,52,56),(9,35,53,55),(10,34,54,60),(11,33,49,59),(12,32,50,58),(13,47,81,67),(14,46,82,72),(15,45,83,71),(16,44,84,70),(17,43,79,69),(18,48,80,68),(19,61,87,41),(20,66,88,40),(21,65,89,39),(22,64,90,38),(23,63,85,37),(24,62,86,42)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 3 | 5 | 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 | 0 | 8 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [8,3,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[9,12,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[7,5,0,0,0,0,6,6,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,12,0,0,0,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | D4⋊2S3 | S3×C4○D4 |
| kernel | C4⋊C4.197D6 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.11D6 | Dic6⋊C4 | Dic3.Q8 | C4⋊C4⋊7S3 | C4⋊C4⋊S3 | C2×C4×Dic3 | C23.28D6 | D4×Dic3 | C23.14D6 | C3×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C2×C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 8 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{197}D_6 % in TeX
G:=Group("C4:C4.197D6"); // GroupNames label
G:=SmallGroup(192,1208);
// by ID
G=gap.SmallGroup(192,1208);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations