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
Subgroups: 476 in 242 conjugacy classes, 100 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×18], Q8 [×6], C23, C23 [×2], C23 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C4○D4 [×12], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×6], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4, C2×C4○D4 [×2], C23.36D4 [×2], C2×C2.D8, C8⋊7D4 [×2], C8.18D4 [×2], C8⋊D4 [×4], C22.31C24 [×2], C2×C8○D4, C2×C4○D8, (C2×C8)⋊14D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8, Q8○D8, (C2×C8)⋊14D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b-1, dcd=c-1 >
►On 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(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 11 51 58)(2 10 52 57)(3 9 53 64)(4 16 54 63)(5 15 55 62)(6 14 56 61)(7 13 49 60)(8 12 50 59)(17 44 27 39)(18 43 28 38)(19 42 29 37)(20 41 30 36)(21 48 31 35)(22 47 32 34)(23 46 25 33)(24 45 26 40)
(1 8)(2 7)(3 6)(4 5)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 24)(18 23)(19 22)(20 21)(25 28)(26 27)(29 32)(30 31)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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), (1,11,51,58)(2,10,52,57)(3,9,53,64)(4,16,54,63)(5,15,55,62)(6,14,56,61)(7,13,49,60)(8,12,50,59)(17,44,27,39)(18,43,28,38)(19,42,29,37)(20,41,30,36)(21,48,31,35)(22,47,32,34)(23,46,25,33)(24,45,26,40), (1,8)(2,7)(3,6)(4,5)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,32)(30,31)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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), (1,11,51,58)(2,10,52,57)(3,9,53,64)(4,16,54,63)(5,15,55,62)(6,14,56,61)(7,13,49,60)(8,12,50,59)(17,44,27,39)(18,43,28,38)(19,42,29,37)(20,41,30,36)(21,48,31,35)(22,47,32,34)(23,46,25,33)(24,45,26,40), (1,8)(2,7)(3,6)(4,5)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,32)(30,31)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(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)], [(1,11,51,58),(2,10,52,57),(3,9,53,64),(4,16,54,63),(5,15,55,62),(6,14,56,61),(7,13,49,60),(8,12,50,59),(17,44,27,39),(18,43,28,38),(19,42,29,37),(20,41,30,36),(21,48,31,35),(22,47,32,34),(23,46,25,33),(24,45,26,40)], [(1,8),(2,7),(3,6),(4,5),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,24),(18,23),(19,22),(20,21),(25,28),(26,27),(29,32),(30,31),(33,43),(34,42),(35,41),(36,48),(37,47),(38,46),(39,45),(40,44),(49,52),(50,51),(53,56),(54,55)])
Matrix representation ►G ⊆ 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] >;
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 |
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