direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8.18D4, C23⋊4Q16, C24.143D4, C8.108(C2×D4), (C2×C8).350D4, C22⋊1(C2×Q16), C4⋊C4.18C23, (C23×C8).15C2, C2.D8⋊45C22, C2.6(C22×Q16), (C2×C4).253C24, (C2×C8).487C23, (C22×Q16)⋊10C2, (C2×Q16)⋊42C22, C4.147(C22×D4), C23.860(C2×D4), (C22×C4).607D4, (C2×Q8).45C23, C4.109(C4⋊D4), Q8⋊C4⋊58C22, C22.92(C4○D8), (C23×C4).702C22, (C22×C8).534C22, C22.513(C22×D4), C22⋊Q8.151C22, C22.173(C4⋊D4), (C22×C4).1532C23, (C22×Q8).278C22, (C2×C2.D8)⋊19C2, C2.15(C2×C4○D8), C4.20(C2×C4○D4), C2.71(C2×C4⋊D4), (C2×Q8⋊C4)⋊18C2, (C2×C4).1425(C2×D4), (C2×C22⋊Q8).52C2, (C2×C4).699(C4○D4), (C2×C4⋊C4).587C22, SmallGroup(128,1781)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 436 in 256 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×26], Q8 [×12], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×10], Q16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×4], C2×Q8 [×10], C24, Q8⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22⋊Q8 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×Q16 [×4], C2×Q16 [×4], C23×C4, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C2.D8, C8.18D4 [×8], C2×C22⋊Q8 [×2], C23×C8, C22×Q16, C2×C8.18D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C8.18D4 [×4], C2×C4⋊D4, C22×Q16, C2×C4○D8, C2×C8.18D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >
►On 64 points
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(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 36 47 61)(2 35 48 60)(3 34 41 59)(4 33 42 58)(5 40 43 57)(6 39 44 64)(7 38 45 63)(8 37 46 62)(9 31 24 53)(10 30 17 52)(11 29 18 51)(12 28 19 50)(13 27 20 49)(14 26 21 56)(15 25 22 55)(16 32 23 54)
(1 57 5 61)(2 64 6 60)(3 63 7 59)(4 62 8 58)(9 49 13 53)(10 56 14 52)(11 55 15 51)(12 54 16 50)(17 26 21 30)(18 25 22 29)(19 32 23 28)(20 31 24 27)(33 42 37 46)(34 41 38 45)(35 48 39 44)(36 47 40 43)
G:=sub<Sym(64)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (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), (1,36,47,61)(2,35,48,60)(3,34,41,59)(4,33,42,58)(5,40,43,57)(6,39,44,64)(7,38,45,63)(8,37,46,62)(9,31,24,53)(10,30,17,52)(11,29,18,51)(12,28,19,50)(13,27,20,49)(14,26,21,56)(15,25,22,55)(16,32,23,54), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,49,13,53)(10,56,14,52)(11,55,15,51)(12,54,16,50)(17,26,21,30)(18,25,22,29)(19,32,23,28)(20,31,24,27)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,45)(18,46)(19,47)(20,48)(21,41)(22,42)(23,43)(24,44)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (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), (1,36,47,61)(2,35,48,60)(3,34,41,59)(4,33,42,58)(5,40,43,57)(6,39,44,64)(7,38,45,63)(8,37,46,62)(9,31,24,53)(10,30,17,52)(11,29,18,51)(12,28,19,50)(13,27,20,49)(14,26,21,56)(15,25,22,55)(16,32,23,54), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,49,13,53)(10,56,14,52)(11,55,15,51)(12,54,16,50)(17,26,21,30)(18,25,22,29)(19,32,23,28)(20,31,24,27)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43) );
G=PermutationGroup([(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,45),(18,46),(19,47),(20,48),(21,41),(22,42),(23,43),(24,44),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(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)], [(1,36,47,61),(2,35,48,60),(3,34,41,59),(4,33,42,58),(5,40,43,57),(6,39,44,64),(7,38,45,63),(8,37,46,62),(9,31,24,53),(10,30,17,52),(11,29,18,51),(12,28,19,50),(13,27,20,49),(14,26,21,56),(15,25,22,55),(16,32,23,54)], [(1,57,5,61),(2,64,6,60),(3,63,7,59),(4,62,8,58),(9,49,13,53),(10,56,14,52),(11,55,15,51),(12,54,16,50),(17,26,21,30),(18,25,22,29),(19,32,23,28),(20,31,24,27),(33,42,37,46),(34,41,38,45),(35,48,39,44),(36,47,40,43)])
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 | 3 | 14 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 9 | 9 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 15 |
| 0 | 0 | 0 | 12 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 15 |
| 0 | 0 | 0 | 13 | 5 |
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],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,9,0,0,0,0,9],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,12,0,0,0,15,5],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,13,0,0,0,15,5] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | Q16 | C4○D8 |
| kernel | C2×C8.18D4 | C2×Q8⋊C4 | C2×C2.D8 | C8.18D4 | C2×C22⋊Q8 | C23×C8 | C22×Q16 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8._{18}D_4 % in TeX
G:=Group("C2xC8.18D4"); // GroupNames label
G:=SmallGroup(128,1781);
// by ID
G=gap.SmallGroup(128,1781);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations