metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.136D6, C6.1132+ (1+4), (C4×Q8)⋊22S3, (C4×D12)⋊42C2, C4⋊C4.303D6, (Q8×C12)⋊20C2, (C4×Dic6)⋊42C2, (C2×Q8).208D6, Dic3⋊5D4⋊18C2, C42⋊7S3⋊21C2, D6.D4⋊10C2, C4.19(C4○D12), C2.25(D4○D12), C4⋊D12.10C2, (C2×C6).129C24, C12.123(C4○D4), C12.23D4⋊10C2, (C2×C12).592C23, (C4×C12).181C22, D6⋊C4.145C22, C4.51(Q8⋊3S3), (C2×D12).29C22, (C6×Q8).229C22, (C22×S3).51C23, C4⋊Dic3.401C22, C22.150(S3×C23), (C2×Dic3).59C23, (C4×Dic3).88C22, Dic3⋊C4.116C22, C3⋊2(C22.53C24), (C2×Dic6).244C22, C6.58(C2×C4○D4), C2.68(C2×C4○D12), (S3×C2×C4).207C22, C2.14(C2×Q8⋊3S3), (C3×C4⋊C4).357C22, (C2×C4).291(C22×S3), SmallGroup(192,1144)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 648 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×9], C22, C22 [×12], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×10], Q8 [×4], C23 [×4], Dic3 [×4], C12 [×4], C12 [×5], D6 [×12], C2×C6, C42, C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×6], C2×Q8, C2×Q8, Dic6 [×2], C4×S3 [×4], D12 [×10], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×4], C4×D4 [×4], C4×Q8, C4×Q8, C22.D4 [×4], C4.4D4 [×4], C4⋊1D4, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×12], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×4], C2×D12 [×6], C6×Q8, C22.53C24, C4×Dic6, C4×D12 [×2], C4⋊D12, C42⋊7S3 [×2], Dic3⋊5D4 [×2], D6.D4 [×4], C12.23D4 [×2], Q8×C12, C42.136D6
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], Q8⋊3S3 [×2], S3×C23, C22.53C24, C2×C4○D12, C2×Q8⋊3S3, D4○D12, C42.136D6
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c6=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=b-1, dcd=a2c5 >
►On 96 points
(1 41 7 47)(2 75 8 81)(3 43 9 37)(4 77 10 83)(5 45 11 39)(6 79 12 73)(13 76 19 82)(14 44 20 38)(15 78 21 84)(16 46 22 40)(17 80 23 74)(18 48 24 42)(25 72 31 66)(26 59 32 53)(27 62 33 68)(28 49 34 55)(29 64 35 70)(30 51 36 57)(50 88 56 94)(52 90 58 96)(54 92 60 86)(61 85 67 91)(63 87 69 93)(65 89 71 95)
(1 58 17 66)(2 59 18 67)(3 60 19 68)(4 49 20 69)(5 50 21 70)(6 51 22 71)(7 52 23 72)(8 53 24 61)(9 54 13 62)(10 55 14 63)(11 56 15 64)(12 57 16 65)(25 41 96 80)(26 42 85 81)(27 43 86 82)(28 44 87 83)(29 45 88 84)(30 46 89 73)(31 47 90 74)(32 48 91 75)(33 37 92 76)(34 38 93 77)(35 39 94 78)(36 40 95 79)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 17)(14 16)(18 24)(19 23)(20 22)(25 27)(28 36)(29 35)(30 34)(31 33)(37 74)(38 73)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)(86 96)(87 95)(88 94)(89 93)(90 92)
G:=sub<Sym(96)| (1,41,7,47)(2,75,8,81)(3,43,9,37)(4,77,10,83)(5,45,11,39)(6,79,12,73)(13,76,19,82)(14,44,20,38)(15,78,21,84)(16,46,22,40)(17,80,23,74)(18,48,24,42)(25,72,31,66)(26,59,32,53)(27,62,33,68)(28,49,34,55)(29,64,35,70)(30,51,36,57)(50,88,56,94)(52,90,58,96)(54,92,60,86)(61,85,67,91)(63,87,69,93)(65,89,71,95), (1,58,17,66)(2,59,18,67)(3,60,19,68)(4,49,20,69)(5,50,21,70)(6,51,22,71)(7,52,23,72)(8,53,24,61)(9,54,13,62)(10,55,14,63)(11,56,15,64)(12,57,16,65)(25,41,96,80)(26,42,85,81)(27,43,86,82)(28,44,87,83)(29,45,88,84)(30,46,89,73)(31,47,90,74)(32,48,91,75)(33,37,92,76)(34,38,93,77)(35,39,94,78)(36,40,95,79), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,74)(38,73)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(86,96)(87,95)(88,94)(89,93)(90,92)>;
G:=Group( (1,41,7,47)(2,75,8,81)(3,43,9,37)(4,77,10,83)(5,45,11,39)(6,79,12,73)(13,76,19,82)(14,44,20,38)(15,78,21,84)(16,46,22,40)(17,80,23,74)(18,48,24,42)(25,72,31,66)(26,59,32,53)(27,62,33,68)(28,49,34,55)(29,64,35,70)(30,51,36,57)(50,88,56,94)(52,90,58,96)(54,92,60,86)(61,85,67,91)(63,87,69,93)(65,89,71,95), (1,58,17,66)(2,59,18,67)(3,60,19,68)(4,49,20,69)(5,50,21,70)(6,51,22,71)(7,52,23,72)(8,53,24,61)(9,54,13,62)(10,55,14,63)(11,56,15,64)(12,57,16,65)(25,41,96,80)(26,42,85,81)(27,43,86,82)(28,44,87,83)(29,45,88,84)(30,46,89,73)(31,47,90,74)(32,48,91,75)(33,37,92,76)(34,38,93,77)(35,39,94,78)(36,40,95,79), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,17)(14,16)(18,24)(19,23)(20,22)(25,27)(28,36)(29,35)(30,34)(31,33)(37,74)(38,73)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)(86,96)(87,95)(88,94)(89,93)(90,92) );
G=PermutationGroup([(1,41,7,47),(2,75,8,81),(3,43,9,37),(4,77,10,83),(5,45,11,39),(6,79,12,73),(13,76,19,82),(14,44,20,38),(15,78,21,84),(16,46,22,40),(17,80,23,74),(18,48,24,42),(25,72,31,66),(26,59,32,53),(27,62,33,68),(28,49,34,55),(29,64,35,70),(30,51,36,57),(50,88,56,94),(52,90,58,96),(54,92,60,86),(61,85,67,91),(63,87,69,93),(65,89,71,95)], [(1,58,17,66),(2,59,18,67),(3,60,19,68),(4,49,20,69),(5,50,21,70),(6,51,22,71),(7,52,23,72),(8,53,24,61),(9,54,13,62),(10,55,14,63),(11,56,15,64),(12,57,16,65),(25,41,96,80),(26,42,85,81),(27,43,86,82),(28,44,87,83),(29,45,88,84),(30,46,89,73),(31,47,90,74),(32,48,91,75),(33,37,92,76),(34,38,93,77),(35,39,94,78),(36,40,95,79)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,17),(14,16),(18,24),(19,23),(20,22),(25,27),(28,36),(29,35),(30,34),(31,33),(37,74),(38,73),(39,84),(40,83),(41,82),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,72),(86,96),(87,95),(88,94),(89,93),(90,92)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 6 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 1 | 7 | 0 | 0 |
| 9 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 1 | 7 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [5,6,0,0,0,8,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,6,10],[1,9,0,0,7,12,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,7,12,0,0,0,0,1,0,0,0,1,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | Q8⋊3S3 | D4○D12 |
| kernel | C42.136D6 | C4×Dic6 | C4×D12 | C4⋊D12 | C42⋊7S3 | Dic3⋊5D4 | D6.D4 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 3 | 1 | 8 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{136}D_6 % in TeX
G:=Group("C4^2.136D6"); // GroupNames label
G:=SmallGroup(192,1144);
// by ID
G=gap.SmallGroup(192,1144);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c^5>;
// generators/relations