direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊8D4, C23⋊5SD16, C24.141D4, C8⋊17(C2×D4), (C2×C8)⋊39D4, (C23×C8)⋊12C2, C4⋊C4.16C23, C22⋊1(C2×SD16), C4.Q8⋊54C22, (C2×C8).485C23, (C2×C4).251C24, (C22×C8)⋊68C22, (C2×D4).56C23, C4.145(C22×D4), (C22×C4).605D4, C23.858(C2×D4), C22⋊Q8⋊65C22, (C2×Q8).44C23, C4.107(C4⋊D4), D4⋊C4⋊56C22, C2.9(C22×SD16), Q8⋊C4⋊57C22, (C22×SD16)⋊25C2, (C2×SD16)⋊75C22, C22.90(C4○D8), C4⋊D4.146C22, (C23×C4).700C22, C22.511(C22×D4), C22.171(C4⋊D4), (C22×C4).1530C23, (C22×D4).344C22, (C22×Q8).277C22, (C2×C4.Q8)⋊27C2, C4.18(C2×C4○D4), C2.13(C2×C4○D8), C2.69(C2×C4⋊D4), (C2×C22⋊Q8)⋊54C2, (C2×D4⋊C4)⋊16C2, (C2×Q8⋊C4)⋊17C2, (C2×C4⋊D4).53C2, (C2×C4).1423(C2×D4), (C2×C4).697(C4○D4), (C2×C4⋊C4).585C22, SmallGroup(128,1779)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 564 in 282 conjugacy classes, 116 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], Q8 [×6], C23, C23 [×6], C23 [×12], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×10], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×2], C2×Q8 [×5], C24, C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C8⋊8D4 [×8], C2×C4⋊D4, C2×C22⋊Q8, C23×C8, C22×SD16, C2×C8⋊8D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×SD16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C8⋊8D4 [×4], C2×C4⋊D4, C22×SD16, C2×C4○D8, C2×C8⋊8D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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 23 45)(2 15 24 48)(3 10 17 43)(4 13 18 46)(5 16 19 41)(6 11 20 44)(7 14 21 47)(8 9 22 42)(25 56 58 40)(26 51 59 35)(27 54 60 38)(28 49 61 33)(29 52 62 36)(30 55 63 39)(31 50 64 34)(32 53 57 37)
(1 39)(2 34)(3 37)(4 40)(5 35)(6 38)(7 33)(8 36)(9 62)(10 57)(11 60)(12 63)(13 58)(14 61)(15 64)(16 59)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,12,23,45)(2,15,24,48)(3,10,17,43)(4,13,18,46)(5,16,19,41)(6,11,20,44)(7,14,21,47)(8,9,22,42)(25,56,58,40)(26,51,59,35)(27,54,60,38)(28,49,61,33)(29,52,62,36)(30,55,63,39)(31,50,64,34)(32,53,57,37), (1,39)(2,34)(3,37)(4,40)(5,35)(6,38)(7,33)(8,36)(9,62)(10,57)(11,60)(12,63)(13,58)(14,61)(15,64)(16,59)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (1,12,23,45)(2,15,24,48)(3,10,17,43)(4,13,18,46)(5,16,19,41)(6,11,20,44)(7,14,21,47)(8,9,22,42)(25,56,58,40)(26,51,59,35)(27,54,60,38)(28,49,61,33)(29,52,62,36)(30,55,63,39)(31,50,64,34)(32,53,57,37), (1,39)(2,34)(3,37)(4,40)(5,35)(6,38)(7,33)(8,36)(9,62)(10,57)(11,60)(12,63)(13,58)(14,61)(15,64)(16,59)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43) );
G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(1,12,23,45),(2,15,24,48),(3,10,17,43),(4,13,18,46),(5,16,19,41),(6,11,20,44),(7,14,21,47),(8,9,22,42),(25,56,58,40),(26,51,59,35),(27,54,60,38),(28,49,61,33),(29,52,62,36),(30,55,63,39),(31,50,64,34),(32,53,57,37)], [(1,39),(2,34),(3,37),(4,40),(5,35),(6,38),(7,33),(8,36),(9,62),(10,57),(11,60),(12,63),(13,58),(14,61),(15,64),(16,59),(17,53),(18,56),(19,51),(20,54),(21,49),(22,52),(23,55),(24,50),(25,46),(26,41),(27,44),(28,47),(29,42),(30,45),(31,48),(32,43)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 5 | 5 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,12,5],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | SD16 | C4○D8 |
| kernel | C2×C8⋊8D4 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.Q8 | C8⋊8D4 | C2×C4⋊D4 | C2×C22⋊Q8 | C23×C8 | C22×SD16 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_8D_4
% in TeX
G:=Group("C2xC8:8D4"); // GroupNames label
G:=SmallGroup(128,1779);
// by ID
G=gap.SmallGroup(128,1779);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations