metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.143D6, C6.1272+ (1+4), (C4×D12)⋊46C2, (Q8×Dic3)⋊20C2, (C4×Dic6)⋊46C2, (D4×Dic3)⋊31C2, (C2×D4).176D6, C4.4D4⋊14S3, C12⋊3D4.9C2, (C2×Q8).163D6, C22⋊C4.36D6, C23.9D6⋊46C2, C2.51(D4○D12), (C2×C6).225C24, D6⋊C4.37C22, Dic3⋊4D4⋊34C2, C12.126(C4○D4), C4.16(D4⋊2S3), C12.23D4⋊23C2, (C2×C12).505C23, (C4×C12).188C22, (C6×D4).158C22, (C22×C6).55C23, C23.57(C22×S3), (C6×Q8).129C22, Dic3.39(C4○D4), C23.11D6⋊41C2, (C2×D12).266C22, C22.D12⋊26C2, (C22×S3).97C23, C4⋊Dic3.235C22, C22.246(S3×C23), Dic3⋊C4.142C22, C3⋊4(C22.53C24), (C4×Dic3).135C22, (C2×Dic3).256C23, (C2×Dic6).250C22, C6.D4.58C22, (C22×Dic3).145C22, C2.81(S3×C4○D4), C6.192(C2×C4○D4), (C3×C4.4D4)⋊17C2, C2.57(C2×D4⋊2S3), (S3×C2×C4).216C22, (C2×C4).198(C22×S3), (C2×C3⋊D4).63C22, (C3×C22⋊C4).67C22, SmallGroup(192,1240)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 608 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×11], C22, C22 [×12], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×4], D6 [×6], C2×C6, C2×C6 [×6], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6 [×2], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊1D4, C4×Dic3 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×4], C6×D4, C6×Q8, C22.53C24, C4×Dic6, C4×D12, Dic3⋊4D4 [×2], C23.9D6 [×2], C23.11D6 [×2], C22.D12 [×2], D4×Dic3, C12⋊3D4, Q8×Dic3, C12.23D4, C3×C4.4D4, C42.143D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), D4⋊2S3 [×2], S3×C23, C22.53C24, C2×D4⋊2S3, S3×C4○D4, D4○D12, C42.143D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
►On 96 points
(1 65 37 83)(2 86 38 70)(3 61 39 79)(4 88 40 72)(5 63 41 81)(6 90 42 68)(7 28 60 49)(8 93 55 76)(9 30 56 51)(10 95 57 78)(11 26 58 53)(12 91 59 74)(13 89 21 67)(14 64 22 82)(15 85 23 69)(16 66 24 84)(17 87 19 71)(18 62 20 80)(25 47 52 35)(27 43 54 31)(29 45 50 33)(32 92 44 75)(34 94 46 77)(36 96 48 73)
(1 12 15 31)(2 60 16 44)(3 8 17 33)(4 56 18 46)(5 10 13 35)(6 58 14 48)(7 24 32 38)(9 20 34 40)(11 22 36 42)(19 45 39 55)(21 47 41 57)(23 43 37 59)(25 63 95 89)(26 82 96 68)(27 65 91 85)(28 84 92 70)(29 61 93 87)(30 80 94 72)(49 66 75 86)(50 79 76 71)(51 62 77 88)(52 81 78 67)(53 64 73 90)(54 83 74 69)
(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 37 42)(2 41 38 5)(3 4 39 40)(7 35 60 47)(8 46 55 34)(9 33 56 45)(10 44 57 32)(11 31 58 43)(12 48 59 36)(13 16 21 24)(14 23 22 15)(17 18 19 20)(25 75 52 92)(26 91 53 74)(27 73 54 96)(28 95 49 78)(29 77 50 94)(30 93 51 76)(61 62 79 80)(63 66 81 84)(64 83 82 65)(67 70 89 86)(68 85 90 69)(71 72 87 88)
G:=sub<Sym(96)| (1,65,37,83)(2,86,38,70)(3,61,39,79)(4,88,40,72)(5,63,41,81)(6,90,42,68)(7,28,60,49)(8,93,55,76)(9,30,56,51)(10,95,57,78)(11,26,58,53)(12,91,59,74)(13,89,21,67)(14,64,22,82)(15,85,23,69)(16,66,24,84)(17,87,19,71)(18,62,20,80)(25,47,52,35)(27,43,54,31)(29,45,50,33)(32,92,44,75)(34,94,46,77)(36,96,48,73), (1,12,15,31)(2,60,16,44)(3,8,17,33)(4,56,18,46)(5,10,13,35)(6,58,14,48)(7,24,32,38)(9,20,34,40)(11,22,36,42)(19,45,39,55)(21,47,41,57)(23,43,37,59)(25,63,95,89)(26,82,96,68)(27,65,91,85)(28,84,92,70)(29,61,93,87)(30,80,94,72)(49,66,75,86)(50,79,76,71)(51,62,77,88)(52,81,78,67)(53,64,73,90)(54,83,74,69), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,35,60,47)(8,46,55,34)(9,33,56,45)(10,44,57,32)(11,31,58,43)(12,48,59,36)(13,16,21,24)(14,23,22,15)(17,18,19,20)(25,75,52,92)(26,91,53,74)(27,73,54,96)(28,95,49,78)(29,77,50,94)(30,93,51,76)(61,62,79,80)(63,66,81,84)(64,83,82,65)(67,70,89,86)(68,85,90,69)(71,72,87,88)>;
G:=Group( (1,65,37,83)(2,86,38,70)(3,61,39,79)(4,88,40,72)(5,63,41,81)(6,90,42,68)(7,28,60,49)(8,93,55,76)(9,30,56,51)(10,95,57,78)(11,26,58,53)(12,91,59,74)(13,89,21,67)(14,64,22,82)(15,85,23,69)(16,66,24,84)(17,87,19,71)(18,62,20,80)(25,47,52,35)(27,43,54,31)(29,45,50,33)(32,92,44,75)(34,94,46,77)(36,96,48,73), (1,12,15,31)(2,60,16,44)(3,8,17,33)(4,56,18,46)(5,10,13,35)(6,58,14,48)(7,24,32,38)(9,20,34,40)(11,22,36,42)(19,45,39,55)(21,47,41,57)(23,43,37,59)(25,63,95,89)(26,82,96,68)(27,65,91,85)(28,84,92,70)(29,61,93,87)(30,80,94,72)(49,66,75,86)(50,79,76,71)(51,62,77,88)(52,81,78,67)(53,64,73,90)(54,83,74,69), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,35,60,47)(8,46,55,34)(9,33,56,45)(10,44,57,32)(11,31,58,43)(12,48,59,36)(13,16,21,24)(14,23,22,15)(17,18,19,20)(25,75,52,92)(26,91,53,74)(27,73,54,96)(28,95,49,78)(29,77,50,94)(30,93,51,76)(61,62,79,80)(63,66,81,84)(64,83,82,65)(67,70,89,86)(68,85,90,69)(71,72,87,88) );
G=PermutationGroup([(1,65,37,83),(2,86,38,70),(3,61,39,79),(4,88,40,72),(5,63,41,81),(6,90,42,68),(7,28,60,49),(8,93,55,76),(9,30,56,51),(10,95,57,78),(11,26,58,53),(12,91,59,74),(13,89,21,67),(14,64,22,82),(15,85,23,69),(16,66,24,84),(17,87,19,71),(18,62,20,80),(25,47,52,35),(27,43,54,31),(29,45,50,33),(32,92,44,75),(34,94,46,77),(36,96,48,73)], [(1,12,15,31),(2,60,16,44),(3,8,17,33),(4,56,18,46),(5,10,13,35),(6,58,14,48),(7,24,32,38),(9,20,34,40),(11,22,36,42),(19,45,39,55),(21,47,41,57),(23,43,37,59),(25,63,95,89),(26,82,96,68),(27,65,91,85),(28,84,92,70),(29,61,93,87),(30,80,94,72),(49,66,75,86),(50,79,76,71),(51,62,77,88),(52,81,78,67),(53,64,73,90),(54,83,74,69)], [(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)], [(1,6,37,42),(2,41,38,5),(3,4,39,40),(7,35,60,47),(8,46,55,34),(9,33,56,45),(10,44,57,32),(11,31,58,43),(12,48,59,36),(13,16,21,24),(14,23,22,15),(17,18,19,20),(25,75,52,92),(26,91,53,74),(27,73,54,96),(28,95,49,78),(29,77,50,94),(30,93,51,76),(61,62,79,80),(63,66,81,84),(64,83,82,65),(67,70,89,86),(68,85,90,69),(71,72,87,88)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 3 | 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 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 8 | 2 | 0 | 0 | 0 | 0 |
| 1 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[12,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[8,1,0,0,0,0,2,5,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,11,8,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2+ (1+4) | D4⋊2S3 | S3×C4○D4 | D4○D12 |
| kernel | C42.143D6 | C4×Dic6 | C4×D12 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | C22.D12 | D4×Dic3 | C12⋊3D4 | Q8×Dic3 | C12.23D4 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C12 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{143}D_6 % in TeX
G:=Group("C4^2.143D6"); // GroupNames label
G:=SmallGroup(192,1240);
// by ID
G=gap.SmallGroup(192,1240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,297,80,6278]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations