metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.160D6, C6.992- 1+4, C12⋊Q8⋊40C2, C4⋊C4.117D6, C42⋊2C2⋊1S3, C42⋊2S3⋊5C2, D6⋊Q8⋊39C2, (C4×Dic6)⋊14C2, C22⋊C4.75D6, (C4×C12).32C22, (C2×C6).246C24, C2.63(Q8○D12), Dic6⋊C4⋊40C2, (C2×C12).192C23, D6⋊C4.139C22, Dic3⋊4D4.4C2, C23.8D6⋊43C2, (C22×C6).60C23, C23.62(C22×S3), Dic3.31(C4○D4), Dic3.D4⋊44C2, C23.16D6⋊20C2, C4⋊Dic3.317C22, C22.267(S3×C23), C23.11D6.4C2, Dic3⋊C4.145C22, (C22×S3).110C23, C3⋊7(C22.50C24), (C2×Dic3).263C23, (C4×Dic3).149C22, (C2×Dic6).254C22, C6.D4.62C22, (C22×Dic3).149C22, C4⋊C4⋊S3⋊39C2, C2.93(S3×C4○D4), C6.204(C2×C4○D4), (C3×C42⋊2C2)⋊1C2, (S3×C2×C4).218C22, (C2×C4).83(C22×S3), (C3×C4⋊C4).201C22, (C2×C3⋊D4).67C22, (C3×C22⋊C4).71C22, SmallGroup(192,1261)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.160D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 480 in 212 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22.50C24, C4×Dic6, C42⋊2S3, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.11D6, Dic6⋊C4, C12⋊Q8, D6⋊Q8, C4⋊C4⋊S3, C3×C42⋊2C2, C42.160D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, S3×C23, C22.50C24, S3×C4○D4, Q8○D12, C42.160D6
►On 96 points
(1 69 57 19)(2 48 58 86)(3 71 59 21)(4 44 60 88)(5 67 55 23)(6 46 56 90)(7 63 25 13)(8 42 26 80)(9 65 27 15)(10 38 28 82)(11 61 29 17)(12 40 30 84)(14 52 64 92)(16 54 66 94)(18 50 62 96)(20 74 70 36)(22 76 72 32)(24 78 68 34)(31 87 75 43)(33 89 77 45)(35 85 73 47)(37 93 81 53)(39 95 83 49)(41 91 79 51)
(1 79 73 13)(2 64 74 42)(3 81 75 15)(4 66 76 38)(5 83 77 17)(6 62 78 40)(7 69 51 47)(8 86 52 20)(9 71 53 43)(10 88 54 22)(11 67 49 45)(12 90 50 24)(14 36 80 58)(16 32 82 60)(18 34 84 56)(19 91 85 25)(21 93 87 27)(23 95 89 29)(26 48 92 70)(28 44 94 72)(30 46 96 68)(31 65 59 37)(33 61 55 39)(35 63 57 41)
(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 18 73 84)(2 17 74 83)(3 16 75 82)(4 15 76 81)(5 14 77 80)(6 13 78 79)(7 68 51 46)(8 67 52 45)(9 72 53 44)(10 71 54 43)(11 70 49 48)(12 69 50 47)(19 96 85 30)(20 95 86 29)(21 94 87 28)(22 93 88 27)(23 92 89 26)(24 91 90 25)(31 38 59 66)(32 37 60 65)(33 42 55 64)(34 41 56 63)(35 40 57 62)(36 39 58 61)
G:=sub<Sym(96)| (1,69,57,19)(2,48,58,86)(3,71,59,21)(4,44,60,88)(5,67,55,23)(6,46,56,90)(7,63,25,13)(8,42,26,80)(9,65,27,15)(10,38,28,82)(11,61,29,17)(12,40,30,84)(14,52,64,92)(16,54,66,94)(18,50,62,96)(20,74,70,36)(22,76,72,32)(24,78,68,34)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,13)(2,64,74,42)(3,81,75,15)(4,66,76,38)(5,83,77,17)(6,62,78,40)(7,69,51,47)(8,86,52,20)(9,71,53,43)(10,88,54,22)(11,67,49,45)(12,90,50,24)(14,36,80,58)(16,32,82,60)(18,34,84,56)(19,91,85,25)(21,93,87,27)(23,95,89,29)(26,48,92,70)(28,44,94,72)(30,46,96,68)(31,65,59,37)(33,61,55,39)(35,63,57,41), (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), (1,18,73,84)(2,17,74,83)(3,16,75,82)(4,15,76,81)(5,14,77,80)(6,13,78,79)(7,68,51,46)(8,67,52,45)(9,72,53,44)(10,71,54,43)(11,70,49,48)(12,69,50,47)(19,96,85,30)(20,95,86,29)(21,94,87,28)(22,93,88,27)(23,92,89,26)(24,91,90,25)(31,38,59,66)(32,37,60,65)(33,42,55,64)(34,41,56,63)(35,40,57,62)(36,39,58,61)>;
G:=Group( (1,69,57,19)(2,48,58,86)(3,71,59,21)(4,44,60,88)(5,67,55,23)(6,46,56,90)(7,63,25,13)(8,42,26,80)(9,65,27,15)(10,38,28,82)(11,61,29,17)(12,40,30,84)(14,52,64,92)(16,54,66,94)(18,50,62,96)(20,74,70,36)(22,76,72,32)(24,78,68,34)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,13)(2,64,74,42)(3,81,75,15)(4,66,76,38)(5,83,77,17)(6,62,78,40)(7,69,51,47)(8,86,52,20)(9,71,53,43)(10,88,54,22)(11,67,49,45)(12,90,50,24)(14,36,80,58)(16,32,82,60)(18,34,84,56)(19,91,85,25)(21,93,87,27)(23,95,89,29)(26,48,92,70)(28,44,94,72)(30,46,96,68)(31,65,59,37)(33,61,55,39)(35,63,57,41), (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), (1,18,73,84)(2,17,74,83)(3,16,75,82)(4,15,76,81)(5,14,77,80)(6,13,78,79)(7,68,51,46)(8,67,52,45)(9,72,53,44)(10,71,54,43)(11,70,49,48)(12,69,50,47)(19,96,85,30)(20,95,86,29)(21,94,87,28)(22,93,88,27)(23,92,89,26)(24,91,90,25)(31,38,59,66)(32,37,60,65)(33,42,55,64)(34,41,56,63)(35,40,57,62)(36,39,58,61) );
G=PermutationGroup([[(1,69,57,19),(2,48,58,86),(3,71,59,21),(4,44,60,88),(5,67,55,23),(6,46,56,90),(7,63,25,13),(8,42,26,80),(9,65,27,15),(10,38,28,82),(11,61,29,17),(12,40,30,84),(14,52,64,92),(16,54,66,94),(18,50,62,96),(20,74,70,36),(22,76,72,32),(24,78,68,34),(31,87,75,43),(33,89,77,45),(35,85,73,47),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(1,79,73,13),(2,64,74,42),(3,81,75,15),(4,66,76,38),(5,83,77,17),(6,62,78,40),(7,69,51,47),(8,86,52,20),(9,71,53,43),(10,88,54,22),(11,67,49,45),(12,90,50,24),(14,36,80,58),(16,32,82,60),(18,34,84,56),(19,91,85,25),(21,93,87,27),(23,95,89,29),(26,48,92,70),(28,44,94,72),(30,46,96,68),(31,65,59,37),(33,61,55,39),(35,63,57,41)], [(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)], [(1,18,73,84),(2,17,74,83),(3,16,75,82),(4,15,76,81),(5,14,77,80),(6,13,78,79),(7,68,51,46),(8,67,52,45),(9,72,53,44),(10,71,54,43),(11,70,49,48),(12,69,50,47),(19,96,85,30),(20,95,86,29),(21,94,87,28),(22,93,88,27),(23,92,89,26),(24,91,90,25),(31,38,59,66),(32,37,60,65),(33,42,55,64),(34,41,56,63),(35,40,57,62),(36,39,58,61)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.160D6 | C4×Dic6 | C42⋊2S3 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.11D6 | Dic6⋊C4 | C12⋊Q8 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 3 | 8 | 1 | 4 | 2 |
Matrix representation of C42.160D6 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,5,0,0,0,0,3,5,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,8,0,0,0,0,10,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C42.160D6 in GAP, Magma, Sage, TeX
C_4^2._{160}D_6 % in TeX
G:=Group("C4^2.160D6"); // GroupNames label
G:=SmallGroup(192,1261);
// by ID
G=gap.SmallGroup(192,1261);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,387,100,794,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations