metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.188D6, C4○D12⋊10C4, D12⋊24(C2×C4), C4⋊C4.309D6, (S3×C42)⋊17C2, Dic6⋊23(C2×C4), (C2×C6).66C24, C6.18(C23×C4), Dic3⋊5D4⋊45C2, D6.5(C22×C4), C42⋊C2⋊20S3, C42⋊2S3⋊29C2, C22⋊C4.126D6, (C22×C4).380D6, Dic3⋊10(C4○D4), Dic6⋊C4⋊44C2, Dic3⋊4D4⋊49C2, C12.143(C22×C4), (C2×C12).877C23, (C4×C12).232C22, D6⋊C4.118C22, C22.28(S3×C23), Dic3.7(C22×C4), (C2×D12).256C22, C23.164(C22×S3), (C22×C6).136C23, Dic3⋊C4.131C22, (C22×S3).163C23, (C22×C12).226C22, (C2×Dic3).306C23, (C2×Dic6).284C22, (C4×Dic3).250C22, (C22×Dic3).216C22, C3⋊2(C4×C4○D4), (C2×C4)⋊10(C4×S3), C3⋊D4⋊6(C2×C4), C4.118(S3×C2×C4), (C2×C4×Dic3)⋊6C2, C2.2(S3×C4○D4), (C4×S3)⋊10(C2×C4), (C2×C12)⋊12(C2×C4), C22.6(S3×C2×C4), C2.20(S3×C22×C4), C6.132(C2×C4○D4), (C3×C42⋊C2)⋊8C2, (C2×C4○D12).17C2, (S3×C2×C4).289C22, (C2×C6).22(C22×C4), (C3×C4⋊C4).305C22, (C2×C4).272(C22×S3), (C2×C3⋊D4).97C22, (C3×C22⋊C4).136C22, SmallGroup(192,1081)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 648 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×14], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×8], Dic3 [×2], C12 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×8], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], C4×S3 [×8], D12 [×4], C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×C6, C2×C42 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C4×Dic3 [×2], C4×Dic3 [×6], Dic3⋊C4 [×4], D6⋊C4 [×4], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×6], C2×D12, C4○D12 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C4×C4○D4, S3×C42 [×2], C42⋊2S3 [×2], Dic3⋊4D4 [×4], Dic6⋊C4 [×2], Dic3⋊5D4 [×2], C2×C4×Dic3, C3×C42⋊C2, C2×C4○D12, C42.188D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C4○D4 [×4], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, C2×C4○D4 [×2], S3×C2×C4 [×6], S3×C23, C4×C4○D4, S3×C22×C4, S3×C4○D4 [×2], C42.188D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
►On 96 points
(1 85 37 55)(2 86 38 56)(3 87 39 57)(4 88 40 58)(5 89 41 59)(6 90 42 60)(7 27 53 78)(8 28 54 73)(9 29 49 74)(10 30 50 75)(11 25 51 76)(12 26 52 77)(13 91 43 61)(14 92 44 62)(15 93 45 63)(16 94 46 64)(17 95 47 65)(18 96 48 66)(19 83 35 67)(20 84 36 68)(21 79 31 69)(22 80 32 70)(23 81 33 71)(24 82 34 72)
(1 7 33 13)(2 54 34 44)(3 9 35 15)(4 50 36 46)(5 11 31 17)(6 52 32 48)(8 24 14 38)(10 20 16 40)(12 22 18 42)(19 45 39 49)(21 47 41 51)(23 43 37 53)(25 69 95 89)(26 80 96 60)(27 71 91 85)(28 82 92 56)(29 67 93 87)(30 84 94 58)(55 78 81 61)(57 74 83 63)(59 76 79 65)(62 86 73 72)(64 88 75 68)(66 90 77 70)
(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 83 37 67)(2 72 38 82)(3 81 39 71)(4 70 40 80)(5 79 41 69)(6 68 42 84)(7 93 53 63)(8 62 54 92)(9 91 49 61)(10 66 50 96)(11 95 51 65)(12 64 52 94)(13 29 43 74)(14 73 44 28)(15 27 45 78)(16 77 46 26)(17 25 47 76)(18 75 48 30)(19 85 35 55)(20 60 36 90)(21 89 31 59)(22 58 32 88)(23 87 33 57)(24 56 34 86)
G:=sub<Sym(96)| (1,85,37,55)(2,86,38,56)(3,87,39,57)(4,88,40,58)(5,89,41,59)(6,90,42,60)(7,27,53,78)(8,28,54,73)(9,29,49,74)(10,30,50,75)(11,25,51,76)(12,26,52,77)(13,91,43,61)(14,92,44,62)(15,93,45,63)(16,94,46,64)(17,95,47,65)(18,96,48,66)(19,83,35,67)(20,84,36,68)(21,79,31,69)(22,80,32,70)(23,81,33,71)(24,82,34,72), (1,7,33,13)(2,54,34,44)(3,9,35,15)(4,50,36,46)(5,11,31,17)(6,52,32,48)(8,24,14,38)(10,20,16,40)(12,22,18,42)(19,45,39,49)(21,47,41,51)(23,43,37,53)(25,69,95,89)(26,80,96,60)(27,71,91,85)(28,82,92,56)(29,67,93,87)(30,84,94,58)(55,78,81,61)(57,74,83,63)(59,76,79,65)(62,86,73,72)(64,88,75,68)(66,90,77,70), (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), (1,83,37,67)(2,72,38,82)(3,81,39,71)(4,70,40,80)(5,79,41,69)(6,68,42,84)(7,93,53,63)(8,62,54,92)(9,91,49,61)(10,66,50,96)(11,95,51,65)(12,64,52,94)(13,29,43,74)(14,73,44,28)(15,27,45,78)(16,77,46,26)(17,25,47,76)(18,75,48,30)(19,85,35,55)(20,60,36,90)(21,89,31,59)(22,58,32,88)(23,87,33,57)(24,56,34,86)>;
G:=Group( (1,85,37,55)(2,86,38,56)(3,87,39,57)(4,88,40,58)(5,89,41,59)(6,90,42,60)(7,27,53,78)(8,28,54,73)(9,29,49,74)(10,30,50,75)(11,25,51,76)(12,26,52,77)(13,91,43,61)(14,92,44,62)(15,93,45,63)(16,94,46,64)(17,95,47,65)(18,96,48,66)(19,83,35,67)(20,84,36,68)(21,79,31,69)(22,80,32,70)(23,81,33,71)(24,82,34,72), (1,7,33,13)(2,54,34,44)(3,9,35,15)(4,50,36,46)(5,11,31,17)(6,52,32,48)(8,24,14,38)(10,20,16,40)(12,22,18,42)(19,45,39,49)(21,47,41,51)(23,43,37,53)(25,69,95,89)(26,80,96,60)(27,71,91,85)(28,82,92,56)(29,67,93,87)(30,84,94,58)(55,78,81,61)(57,74,83,63)(59,76,79,65)(62,86,73,72)(64,88,75,68)(66,90,77,70), (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), (1,83,37,67)(2,72,38,82)(3,81,39,71)(4,70,40,80)(5,79,41,69)(6,68,42,84)(7,93,53,63)(8,62,54,92)(9,91,49,61)(10,66,50,96)(11,95,51,65)(12,64,52,94)(13,29,43,74)(14,73,44,28)(15,27,45,78)(16,77,46,26)(17,25,47,76)(18,75,48,30)(19,85,35,55)(20,60,36,90)(21,89,31,59)(22,58,32,88)(23,87,33,57)(24,56,34,86) );
G=PermutationGroup([(1,85,37,55),(2,86,38,56),(3,87,39,57),(4,88,40,58),(5,89,41,59),(6,90,42,60),(7,27,53,78),(8,28,54,73),(9,29,49,74),(10,30,50,75),(11,25,51,76),(12,26,52,77),(13,91,43,61),(14,92,44,62),(15,93,45,63),(16,94,46,64),(17,95,47,65),(18,96,48,66),(19,83,35,67),(20,84,36,68),(21,79,31,69),(22,80,32,70),(23,81,33,71),(24,82,34,72)], [(1,7,33,13),(2,54,34,44),(3,9,35,15),(4,50,36,46),(5,11,31,17),(6,52,32,48),(8,24,14,38),(10,20,16,40),(12,22,18,42),(19,45,39,49),(21,47,41,51),(23,43,37,53),(25,69,95,89),(26,80,96,60),(27,71,91,85),(28,82,92,56),(29,67,93,87),(30,84,94,58),(55,78,81,61),(57,74,83,63),(59,76,79,65),(62,86,73,72),(64,88,75,68),(66,90,77,70)], [(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)], [(1,83,37,67),(2,72,38,82),(3,81,39,71),(4,70,40,80),(5,79,41,69),(6,68,42,84),(7,93,53,63),(8,62,54,92),(9,91,49,61),(10,66,50,96),(11,95,51,65),(12,64,52,94),(13,29,43,74),(14,73,44,28),(15,27,45,78),(16,77,46,26),(17,25,47,76),(18,75,48,30),(19,85,35,55),(20,60,36,90),(21,89,31,59),(22,58,32,88),(23,87,33,57),(24,56,34,86)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0],[0,12,0,0,1,1,0,0,0,0,0,12,0,0,12,0],[1,1,0,0,0,12,0,0,0,0,8,0,0,0,0,5] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 4W | ··· | 4AD | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4×S3 | S3×C4○D4 |
| kernel | C42.188D6 | S3×C42 | C42⋊2S3 | Dic3⋊4D4 | Dic6⋊C4 | Dic3⋊5D4 | C2×C4×Dic3 | C3×C42⋊C2 | C2×C4○D12 | C4○D12 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{188}D_6 % in TeX
G:=Group("C4^2.188D6"); // GroupNames label
G:=SmallGroup(192,1081);
// by ID
G=gap.SmallGroup(192,1081);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations