metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.105D6, C6.562- (1+4), C4⋊C4.280D6, (C4×D4).12S3, (C4×Dic6)⋊27C2, (D4×C12).13C2, (C2×D4).209D6, Dic3.Q8⋊7C2, (C2×C6).85C24, C12.48D4⋊8C2, C2.14(Q8○D12), C22⋊C4.130D6, C12.6Q8⋊15C2, (C22×C4).220D6, C23.8D6⋊6C2, C12.291(C4○D4), (C4×C12).147C22, (C2×C12).585C23, (C6×D4).303C22, C23.26D6⋊6C2, C4.115(D4⋊2S3), C22.11(C4○D12), C23.16D6⋊27C2, C4⋊Dic3.296C22, C22.113(S3×C23), (C22×C12).79C22, (C22×C6).155C23, C23.175(C22×S3), (C2×Dic3).35C23, C6.D4.9C22, C23.23D6.5C2, Dic3⋊C4.153C22, C3⋊4(C22.46C24), (C2×Dic6).236C22, (C4×Dic3).202C22, (C22×Dic3).93C22, C6.37(C2×C4○D4), (C2×C4⋊Dic3)⋊23C2, C2.41(C2×C4○D12), (C2×C6).15(C4○D4), C2.19(C2×D4⋊2S3), (C3×C4⋊C4).321C22, (C2×C4).155(C22×S3), (C3×C22⋊C4).142C22, SmallGroup(192,1100)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 214 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×8], C12 [×2], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8, Dic6 [×2], C2×Dic3 [×8], C2×Dic3 [×4], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×C6 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×Dic3 [×4], Dic3⋊C4 [×10], C4⋊Dic3 [×3], C4⋊Dic3 [×2], C6.D4 [×6], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×Dic3 [×2], C22×C12 [×2], C6×D4, C22.46C24, C4×Dic6, C12.6Q8, C23.16D6 [×2], C23.8D6 [×2], Dic3.Q8 [×2], C12.48D4 [×2], C2×C4⋊Dic3, C23.26D6, C23.23D6 [×2], D4×C12, C42.105D6
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), C4○D12 [×2], D4⋊2S3 [×2], S3×C23, C22.46C24, C2×C4○D12, C2×D4⋊2S3, Q8○D12, C42.105D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
►On 96 points
(1 40 37 4)(2 5 38 41)(3 42 39 6)(7 51 54 10)(8 11 49 52)(9 53 50 12)(13 46 43 16)(14 17 44 47)(15 48 45 18)(19 32 35 22)(20 23 36 33)(21 34 31 24)(25 65 78 92)(26 93 73 66)(27 61 74 94)(28 95 75 62)(29 63 76 96)(30 91 77 64)(55 83 88 72)(56 67 89 84)(57 79 90 68)(58 69 85 80)(59 81 86 70)(60 71 87 82)
(1 7 33 13)(2 8 34 14)(3 9 35 15)(4 10 36 16)(5 11 31 17)(6 12 32 18)(19 45 39 50)(20 46 40 51)(21 47 41 52)(22 48 42 53)(23 43 37 54)(24 44 38 49)(25 89 95 70)(26 90 96 71)(27 85 91 72)(28 86 92 67)(29 87 93 68)(30 88 94 69)(55 61 80 77)(56 62 81 78)(57 63 82 73)(58 64 83 74)(59 65 84 75)(60 66 79 76)
(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 88 37 55)(2 87 38 60)(3 86 39 59)(4 85 40 58)(5 90 41 57)(6 89 42 56)(7 30 54 77)(8 29 49 76)(9 28 50 75)(10 27 51 74)(11 26 52 73)(12 25 53 78)(13 94 43 61)(14 93 44 66)(15 92 45 65)(16 91 46 64)(17 96 47 63)(18 95 48 62)(19 84 35 67)(20 83 36 72)(21 82 31 71)(22 81 32 70)(23 80 33 69)(24 79 34 68)
G:=sub<Sym(96)| (1,40,37,4)(2,5,38,41)(3,42,39,6)(7,51,54,10)(8,11,49,52)(9,53,50,12)(13,46,43,16)(14,17,44,47)(15,48,45,18)(19,32,35,22)(20,23,36,33)(21,34,31,24)(25,65,78,92)(26,93,73,66)(27,61,74,94)(28,95,75,62)(29,63,76,96)(30,91,77,64)(55,83,88,72)(56,67,89,84)(57,79,90,68)(58,69,85,80)(59,81,86,70)(60,71,87,82), (1,7,33,13)(2,8,34,14)(3,9,35,15)(4,10,36,16)(5,11,31,17)(6,12,32,18)(19,45,39,50)(20,46,40,51)(21,47,41,52)(22,48,42,53)(23,43,37,54)(24,44,38,49)(25,89,95,70)(26,90,96,71)(27,85,91,72)(28,86,92,67)(29,87,93,68)(30,88,94,69)(55,61,80,77)(56,62,81,78)(57,63,82,73)(58,64,83,74)(59,65,84,75)(60,66,79,76), (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), (1,88,37,55)(2,87,38,60)(3,86,39,59)(4,85,40,58)(5,90,41,57)(6,89,42,56)(7,30,54,77)(8,29,49,76)(9,28,50,75)(10,27,51,74)(11,26,52,73)(12,25,53,78)(13,94,43,61)(14,93,44,66)(15,92,45,65)(16,91,46,64)(17,96,47,63)(18,95,48,62)(19,84,35,67)(20,83,36,72)(21,82,31,71)(22,81,32,70)(23,80,33,69)(24,79,34,68)>;
G:=Group( (1,40,37,4)(2,5,38,41)(3,42,39,6)(7,51,54,10)(8,11,49,52)(9,53,50,12)(13,46,43,16)(14,17,44,47)(15,48,45,18)(19,32,35,22)(20,23,36,33)(21,34,31,24)(25,65,78,92)(26,93,73,66)(27,61,74,94)(28,95,75,62)(29,63,76,96)(30,91,77,64)(55,83,88,72)(56,67,89,84)(57,79,90,68)(58,69,85,80)(59,81,86,70)(60,71,87,82), (1,7,33,13)(2,8,34,14)(3,9,35,15)(4,10,36,16)(5,11,31,17)(6,12,32,18)(19,45,39,50)(20,46,40,51)(21,47,41,52)(22,48,42,53)(23,43,37,54)(24,44,38,49)(25,89,95,70)(26,90,96,71)(27,85,91,72)(28,86,92,67)(29,87,93,68)(30,88,94,69)(55,61,80,77)(56,62,81,78)(57,63,82,73)(58,64,83,74)(59,65,84,75)(60,66,79,76), (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), (1,88,37,55)(2,87,38,60)(3,86,39,59)(4,85,40,58)(5,90,41,57)(6,89,42,56)(7,30,54,77)(8,29,49,76)(9,28,50,75)(10,27,51,74)(11,26,52,73)(12,25,53,78)(13,94,43,61)(14,93,44,66)(15,92,45,65)(16,91,46,64)(17,96,47,63)(18,95,48,62)(19,84,35,67)(20,83,36,72)(21,82,31,71)(22,81,32,70)(23,80,33,69)(24,79,34,68) );
G=PermutationGroup([(1,40,37,4),(2,5,38,41),(3,42,39,6),(7,51,54,10),(8,11,49,52),(9,53,50,12),(13,46,43,16),(14,17,44,47),(15,48,45,18),(19,32,35,22),(20,23,36,33),(21,34,31,24),(25,65,78,92),(26,93,73,66),(27,61,74,94),(28,95,75,62),(29,63,76,96),(30,91,77,64),(55,83,88,72),(56,67,89,84),(57,79,90,68),(58,69,85,80),(59,81,86,70),(60,71,87,82)], [(1,7,33,13),(2,8,34,14),(3,9,35,15),(4,10,36,16),(5,11,31,17),(6,12,32,18),(19,45,39,50),(20,46,40,51),(21,47,41,52),(22,48,42,53),(23,43,37,54),(24,44,38,49),(25,89,95,70),(26,90,96,71),(27,85,91,72),(28,86,92,67),(29,87,93,68),(30,88,94,69),(55,61,80,77),(56,62,81,78),(57,63,82,73),(58,64,83,74),(59,65,84,75),(60,66,79,76)], [(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)], [(1,88,37,55),(2,87,38,60),(3,86,39,59),(4,85,40,58),(5,90,41,57),(6,89,42,56),(7,30,54,77),(8,29,49,76),(9,28,50,75),(10,27,51,74),(11,26,52,73),(12,25,53,78),(13,94,43,61),(14,93,44,66),(15,92,45,65),(16,91,46,64),(17,96,47,63),(18,95,48,62),(19,84,35,67),(20,83,36,72),(21,82,31,71),(22,81,32,70),(23,80,33,69),(24,79,34,68)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 10 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 8 | 2 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 4 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 9 | 0 | 0 |
| 7 | 9 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,10,12,0,0,0,0,0,12,0,0,1,0],[8,0,0,0,2,5,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,4,3,0,0,0,0,0,1,0,0,1,0],[4,7,0,0,9,9,0,0,0,0,0,8,0,0,8,0] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- (1+4) | D4⋊2S3 | Q8○D12 |
| kernel | C42.105D6 | C4×Dic6 | C12.6Q8 | C23.16D6 | C23.8D6 | Dic3.Q8 | C12.48D4 | C2×C4⋊Dic3 | C23.26D6 | C23.23D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C22 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{105}D_6 % in TeX
G:=Group("C4^2.105D6"); // GroupNames label
G:=SmallGroup(192,1100);
// by ID
G=gap.SmallGroup(192,1100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,100,675,570,6278]);
// 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=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations