p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4:6M4(2), C42.295C23, (C8xD4):44C2, D4o2(C8:C4), C8:16(C4oD4), C8:9D4:38C2, C8:4Q8:38C2, (C4xD4).30C4, (C4xQ8).28C4, C4:C8.235C22, (C4xM4(2)):38C2, (C2xC8).430C23, (C4xC8).336C22, (C2xC4).667C24, C42.218(C2xC4), C4.33(C2xM4(2)), C42:C2.32C4, C42.6C4:48C2, (C4xD4).296C22, (C4xQ8).281C22, C8:C4.176C22, C22.6(C2xM4(2)), C2.25(Q8oM4(2)), C22:C8.143C22, (C2xC42).777C22, C22.192(C23xC4), C23.148(C22xC4), (C22xC8).449C22, (C22xC4).937C23, C2.16(C22xM4(2)), (C2xM4(2)).369C22, C8:C4o(C4xD4), C4:C4o(C8:C4), (C2xD4)o(C8:C4), (C2xC8:C4):36C2, C2.49(C4xC4oD4), C4:C4.226(C2xC4), C22:C4o(C8:C4), (C2xC4oD4).26C4, (C4xC4oD4).16C2, C4.318(C2xC4oD4), (C2xD4).251(C2xC4), C22:C4.92(C2xC4), (C2xQ8).210(C2xC4), (C2xC4).297(C22xC4), (C22xC4).351(C2xC4), SmallGroup(128,1702)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4:6M4(2)
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c5 >
Subgroups: 276 in 203 conjugacy classes, 138 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4xC8, C8:C4, C8:C4, C22:C8, C4:C8, C2xC42, C2xC42, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22xC8, C2xM4(2), C2xC4oD4, C2xC8:C4, C4xM4(2), C42.6C4, C42.6C4, C8xD4, C8:9D4, C8:4Q8, C4xC4oD4, D4:6M4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, M4(2), C22xC4, C4oD4, C24, C2xM4(2), C23xC4, C2xC4oD4, C4xC4oD4, C22xM4(2), Q8oM4(2), D4:6M4(2)
►On 64 points
(1 10 27 39)(2 11 28 40)(3 12 29 33)(4 13 30 34)(5 14 31 35)(6 15 32 36)(7 16 25 37)(8 9 26 38)(17 62 41 53)(18 63 42 54)(19 64 43 55)(20 57 44 56)(21 58 45 49)(22 59 46 50)(23 60 47 51)(24 61 48 52)
(1 18)(2 43)(3 20)(4 45)(5 22)(6 47)(7 24)(8 41)(9 62)(10 54)(11 64)(12 56)(13 58)(14 50)(15 60)(16 52)(17 26)(19 28)(21 30)(23 32)(25 48)(27 42)(29 44)(31 46)(33 57)(34 49)(35 59)(36 51)(37 61)(38 53)(39 63)(40 55)
(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)
(2 6)(4 8)(9 13)(11 15)(17 45)(18 42)(19 47)(20 44)(21 41)(22 46)(23 43)(24 48)(26 30)(28 32)(34 38)(36 40)(49 62)(50 59)(51 64)(52 61)(53 58)(54 63)(55 60)(56 57)
G:=sub<Sym(64)| (1,10,27,39)(2,11,28,40)(3,12,29,33)(4,13,30,34)(5,14,31,35)(6,15,32,36)(7,16,25,37)(8,9,26,38)(17,62,41,53)(18,63,42,54)(19,64,43,55)(20,57,44,56)(21,58,45,49)(22,59,46,50)(23,60,47,51)(24,61,48,52), (1,18)(2,43)(3,20)(4,45)(5,22)(6,47)(7,24)(8,41)(9,62)(10,54)(11,64)(12,56)(13,58)(14,50)(15,60)(16,52)(17,26)(19,28)(21,30)(23,32)(25,48)(27,42)(29,44)(31,46)(33,57)(34,49)(35,59)(36,51)(37,61)(38,53)(39,63)(40,55), (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), (2,6)(4,8)(9,13)(11,15)(17,45)(18,42)(19,47)(20,44)(21,41)(22,46)(23,43)(24,48)(26,30)(28,32)(34,38)(36,40)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57)>;
G:=Group( (1,10,27,39)(2,11,28,40)(3,12,29,33)(4,13,30,34)(5,14,31,35)(6,15,32,36)(7,16,25,37)(8,9,26,38)(17,62,41,53)(18,63,42,54)(19,64,43,55)(20,57,44,56)(21,58,45,49)(22,59,46,50)(23,60,47,51)(24,61,48,52), (1,18)(2,43)(3,20)(4,45)(5,22)(6,47)(7,24)(8,41)(9,62)(10,54)(11,64)(12,56)(13,58)(14,50)(15,60)(16,52)(17,26)(19,28)(21,30)(23,32)(25,48)(27,42)(29,44)(31,46)(33,57)(34,49)(35,59)(36,51)(37,61)(38,53)(39,63)(40,55), (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), (2,6)(4,8)(9,13)(11,15)(17,45)(18,42)(19,47)(20,44)(21,41)(22,46)(23,43)(24,48)(26,30)(28,32)(34,38)(36,40)(49,62)(50,59)(51,64)(52,61)(53,58)(54,63)(55,60)(56,57) );
G=PermutationGroup([[(1,10,27,39),(2,11,28,40),(3,12,29,33),(4,13,30,34),(5,14,31,35),(6,15,32,36),(7,16,25,37),(8,9,26,38),(17,62,41,53),(18,63,42,54),(19,64,43,55),(20,57,44,56),(21,58,45,49),(22,59,46,50),(23,60,47,51),(24,61,48,52)], [(1,18),(2,43),(3,20),(4,45),(5,22),(6,47),(7,24),(8,41),(9,62),(10,54),(11,64),(12,56),(13,58),(14,50),(15,60),(16,52),(17,26),(19,28),(21,30),(23,32),(25,48),(27,42),(29,44),(31,46),(33,57),(34,49),(35,59),(36,51),(37,61),(38,53),(39,63),(40,55)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,45),(18,42),(19,47),(20,44),(21,41),(22,46),(23,43),(24,48),(26,30),(28,32),(34,38),(36,40),(49,62),(50,59),(51,64),(52,61),(53,58),(54,63),(55,60),(56,57)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4U | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4oD4 | M4(2) | Q8oM4(2) |
| kernel | D4:6M4(2) | C2xC8:C4 | C4xM4(2) | C42.6C4 | C8xD4 | C8:9D4 | C8:4Q8 | C4xC4oD4 | C42:C2 | C4xD4 | C4xQ8 | C2xC4oD4 | C8 | D4 | C2 |
| # reps | 1 | 2 | 1 | 3 | 2 | 4 | 2 | 1 | 6 | 6 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of D4:6M4(2) ►in GL4(F17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 14 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 3 | 8 |
| 0 | 0 | 16 | 14 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 3 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,14,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,3,16,0,0,8,14],[0,4,0,0,1,0,0,0,0,0,13,3,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,1,12,0,0,0,16] >;
D4:6M4(2) in GAP, Magma, Sage, TeX
D_4\rtimes_6M_4(2)
% in TeX
G:=Group("D4:6M4(2)"); // GroupNames label
G:=SmallGroup(128,1702);
// by ID
G=gap.SmallGroup(128,1702);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,80,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^5>;
// generators/relations