metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.160D6, C6.992- (1+4), C12⋊Q8⋊40C2, C4⋊C4.117D6, C42⋊2S3⋊5C2, C42⋊2C2⋊1S3, 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, C23.62(C22×S3), (C22×C6).60C23, 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×Dic6).254C22, (C2×Dic3).263C23, (C4×Dic3).149C22, 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
Subgroups: 480 in 212 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×15], C22, C22 [×6], S3, C6 [×3], C6, C2×C4 [×6], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, Dic3 [×4], Dic3 [×5], C12 [×6], D6 [×3], C2×C6, C2×C6 [×3], C42, C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic6 [×6], C4×S3 [×2], C2×Dic3 [×7], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×6], C22×S3, C22×C6, C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8, C4×Dic3 [×6], Dic3⋊C4 [×7], C4⋊Dic3 [×2], D6⋊C4 [×5], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22.50C24, C4×Dic6, C42⋊2S3, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.11D6 [×2], Dic6⋊C4 [×2], C12⋊Q8, D6⋊Q8, C4⋊C4⋊S3 [×2], C3×C42⋊2C2, C42.160D6
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), S3×C23, C22.50C24, S3×C4○D4 [×2], Q8○D12, C42.160D6
Generators and relations
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 >
►On 96 points
(1 67 55 23)(2 48 56 86)(3 69 57 19)(4 44 58 88)(5 71 59 21)(6 46 60 90)(7 61 29 17)(8 42 30 80)(9 63 25 13)(10 38 26 82)(11 65 27 15)(12 40 28 84)(14 54 64 94)(16 50 66 96)(18 52 62 92)(20 76 70 32)(22 78 72 34)(24 74 68 36)(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 17)(2 62 74 42)(3 81 75 13)(4 64 76 38)(5 83 77 15)(6 66 78 40)(7 67 51 47)(8 86 52 24)(9 69 53 43)(10 88 54 20)(11 71 49 45)(12 90 50 22)(14 32 82 58)(16 34 84 60)(18 36 80 56)(19 93 87 25)(21 95 89 27)(23 91 85 29)(26 44 94 70)(28 46 96 72)(30 48 92 68)(31 63 57 37)(33 65 59 39)(35 61 55 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 16 73 84)(2 15 74 83)(3 14 75 82)(4 13 76 81)(5 18 77 80)(6 17 78 79)(7 72 51 46)(8 71 52 45)(9 70 53 44)(10 69 54 43)(11 68 49 48)(12 67 50 47)(19 94 87 26)(20 93 88 25)(21 92 89 30)(22 91 90 29)(23 96 85 28)(24 95 86 27)(31 38 57 64)(32 37 58 63)(33 42 59 62)(34 41 60 61)(35 40 55 66)(36 39 56 65)
G:=sub<Sym(96)| (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,62,74,42)(3,81,75,13)(4,64,76,38)(5,83,77,15)(6,66,78,40)(7,67,51,47)(8,86,52,24)(9,69,53,43)(10,88,54,20)(11,71,49,45)(12,90,50,22)(14,32,82,58)(16,34,84,60)(18,36,80,56)(19,93,87,25)(21,95,89,27)(23,91,85,29)(26,44,94,70)(28,46,96,72)(30,48,92,68)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,16,73,84)(2,15,74,83)(3,14,75,82)(4,13,76,81)(5,18,77,80)(6,17,78,79)(7,72,51,46)(8,71,52,45)(9,70,53,44)(10,69,54,43)(11,68,49,48)(12,67,50,47)(19,94,87,26)(20,93,88,25)(21,92,89,30)(22,91,90,29)(23,96,85,28)(24,95,86,27)(31,38,57,64)(32,37,58,63)(33,42,59,62)(34,41,60,61)(35,40,55,66)(36,39,56,65)>;
G:=Group( (1,67,55,23)(2,48,56,86)(3,69,57,19)(4,44,58,88)(5,71,59,21)(6,46,60,90)(7,61,29,17)(8,42,30,80)(9,63,25,13)(10,38,26,82)(11,65,27,15)(12,40,28,84)(14,54,64,94)(16,50,66,96)(18,52,62,92)(20,76,70,32)(22,78,72,34)(24,74,68,36)(31,87,75,43)(33,89,77,45)(35,85,73,47)(37,93,81,53)(39,95,83,49)(41,91,79,51), (1,79,73,17)(2,62,74,42)(3,81,75,13)(4,64,76,38)(5,83,77,15)(6,66,78,40)(7,67,51,47)(8,86,52,24)(9,69,53,43)(10,88,54,20)(11,71,49,45)(12,90,50,22)(14,32,82,58)(16,34,84,60)(18,36,80,56)(19,93,87,25)(21,95,89,27)(23,91,85,29)(26,44,94,70)(28,46,96,72)(30,48,92,68)(31,63,57,37)(33,65,59,39)(35,61,55,41), (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), (1,16,73,84)(2,15,74,83)(3,14,75,82)(4,13,76,81)(5,18,77,80)(6,17,78,79)(7,72,51,46)(8,71,52,45)(9,70,53,44)(10,69,54,43)(11,68,49,48)(12,67,50,47)(19,94,87,26)(20,93,88,25)(21,92,89,30)(22,91,90,29)(23,96,85,28)(24,95,86,27)(31,38,57,64)(32,37,58,63)(33,42,59,62)(34,41,60,61)(35,40,55,66)(36,39,56,65) );
G=PermutationGroup([(1,67,55,23),(2,48,56,86),(3,69,57,19),(4,44,58,88),(5,71,59,21),(6,46,60,90),(7,61,29,17),(8,42,30,80),(9,63,25,13),(10,38,26,82),(11,65,27,15),(12,40,28,84),(14,54,64,94),(16,50,66,96),(18,52,62,92),(20,76,70,32),(22,78,72,34),(24,74,68,36),(31,87,75,43),(33,89,77,45),(35,85,73,47),(37,93,81,53),(39,95,83,49),(41,91,79,51)], [(1,79,73,17),(2,62,74,42),(3,81,75,13),(4,64,76,38),(5,83,77,15),(6,66,78,40),(7,67,51,47),(8,86,52,24),(9,69,53,43),(10,88,54,20),(11,71,49,45),(12,90,50,22),(14,32,82,58),(16,34,84,60),(18,36,80,56),(19,93,87,25),(21,95,89,27),(23,91,85,29),(26,44,94,70),(28,46,96,72),(30,48,92,68),(31,63,57,37),(33,65,59,39),(35,61,55,41)], [(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)], [(1,16,73,84),(2,15,74,83),(3,14,75,82),(4,13,76,81),(5,18,77,80),(6,17,78,79),(7,72,51,46),(8,71,52,45),(9,70,53,44),(10,69,54,43),(11,68,49,48),(12,67,50,47),(19,94,87,26),(20,93,88,25),(21,92,89,30),(22,91,90,29),(23,96,85,28),(24,95,86,27),(31,38,57,64),(32,37,58,63),(33,42,59,62),(34,41,60,61),(35,40,55,66),(36,39,56,65)])
Matrix representation ►G ⊆ 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] >;
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 |
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