p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊14D4, C8⋊D4⋊4C2, C8⋊7D4⋊35C2, C8.111(C2×D4), (C2×D4).220D4, C2.11(D4○D8), C4⋊C4.30C23, (C2×Q8).175D4, C2.11(Q8○D8), C23.79(C2×D4), C8.18D4⋊35C2, (C2×C4).265C24, (C2×C8).257C23, (C2×D4).67C23, C4.159(C22×D4), (C2×Q8).55C23, C4.175(C4⋊D4), (C2×D8).120C22, C4⋊D4.21C22, C2.D8.163C22, C22⋊Q8.21C22, C23.36D4⋊43C2, C22.15(C4⋊D4), (C22×C4).987C23, (C22×C8).262C22, (C2×Q16).118C22, C22.525(C22×D4), D4⋊C4.131C22, C22.31C24⋊6C2, Q8⋊C4.124C22, (C2×SD16).113C22, (C2×M4(2)).267C22, (C2×C8○D4)⋊4C2, (C2×C4○D8)⋊18C2, (C2×C2.D8)⋊41C2, C4.32(C2×C4○D4), (C2×C4).133(C2×D4), C2.83(C2×C4⋊D4), (C2×C4).286(C4○D4), (C2×C4⋊C4).594C22, (C2×C4○D4).128C22, SmallGroup(128,1793)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8)⋊14D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 476 in 242 conjugacy classes, 100 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, C23.36D4, C2×C2.D8, C8⋊7D4, C8.18D4, C8⋊D4, C22.31C24, C2×C8○D4, C2×C4○D8, (C2×C8)⋊14D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○D8, Q8○D8, (C2×C8)⋊14D4
►On 64 points
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(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)
(1 12 51 58)(2 11 52 57)(3 10 53 64)(4 9 54 63)(5 16 55 62)(6 15 56 61)(7 14 49 60)(8 13 50 59)(17 43 29 37)(18 42 30 36)(19 41 31 35)(20 48 32 34)(21 47 25 33)(22 46 26 40)(23 45 27 39)(24 44 28 38)
(1 8)(2 7)(3 6)(4 5)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 64)(16 63)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(64)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (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), (1,12,51,58)(2,11,52,57)(3,10,53,64)(4,9,54,63)(5,16,55,62)(6,15,56,61)(7,14,49,60)(8,13,50,59)(17,43,29,37)(18,42,30,36)(19,41,31,35)(20,48,32,34)(21,47,25,33)(22,46,26,40)(23,45,27,39)(24,44,28,38), (1,8)(2,7)(3,6)(4,5)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,64)(16,63)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (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), (1,12,51,58)(2,11,52,57)(3,10,53,64)(4,9,54,63)(5,16,55,62)(6,15,56,61)(7,14,49,60)(8,13,50,59)(17,43,29,37)(18,42,30,36)(19,41,31,35)(20,48,32,34)(21,47,25,33)(22,46,26,40)(23,45,27,39)(24,44,28,38), (1,8)(2,7)(3,6)(4,5)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,64)(16,63)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61)], [(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)], [(1,12,51,58),(2,11,52,57),(3,10,53,64),(4,9,54,63),(5,16,55,62),(6,15,56,61),(7,14,49,60),(8,13,50,59),(17,43,29,37),(18,42,30,36),(19,41,31,35),(20,48,32,34),(21,47,25,33),(22,46,26,40),(23,45,27,39),(24,44,28,38)], [(1,8),(2,7),(3,6),(4,5),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,64),(16,63),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,44),(34,43),(35,42),(36,41),(37,48),(38,47),(39,46),(40,45),(49,52),(50,51),(53,56),(54,55)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D4○D8 | Q8○D8 |
| kernel | (C2×C8)⋊14D4 | C23.36D4 | C2×C2.D8 | C8⋊7D4 | C8.18D4 | C8⋊D4 | C22.31C24 | C2×C8○D4 | C2×C4○D8 | C2×C8 | C2×D4 | C2×Q8 | C2×C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 2 | 2 |
Matrix representation of (C2×C8)⋊14D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 2 | 16 | 0 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 1 | 16 |
| 0 | 0 | 12 | 12 | 16 | 16 |
| 0 | 0 | 1 | 16 | 12 | 5 |
| 0 | 0 | 16 | 16 | 5 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[2,5,0,0,0,0,16,15,0,0,0,0,0,0,5,12,1,16,0,0,12,12,16,16,0,0,1,16,12,5,0,0,16,16,5,5],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14] >;
(C2×C8)⋊14D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_{14}D_4 % in TeX
G:=Group("(C2xC8):14D4"); // GroupNames label
G:=SmallGroup(128,1793);
// by ID
G=gap.SmallGroup(128,1793);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^4,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations