p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4).57D8, C2.10(C4×D8), D4⋊C4⋊1C4, (C2×C8).332D4, C4.123(C4×D4), C2.3(C8⋊7D4), C2.5(C8⋊8D4), C2.17(C4×SD16), (C2×C4).62SD16, C22.43(C2×D8), C2.2(C4.4D8), C23.785(C2×D4), C22.168(C4×D4), (C22×C4).554D4, C22.4Q16⋊18C2, C4.8(C42⋊C2), C4.2(C42⋊2C2), C22.70(C4○D8), C22.67(C2×SD16), (C22×C8).484C22, (C22×D4).39C22, C22.131(C4⋊D4), C23.65C23⋊3C2, (C22×C4).1385C23, (C2×C42).1065C22, C22.59(C4.4D4), C4.93(C22.D4), C24.3C22.8C2, C2.3(C42.78C22), C2.19(C24.C22), (C2×C4×C8)⋊10C2, C4⋊C4.82(C2×C4), (C2×C8).160(C2×C4), (C2×D4).99(C2×C4), (C2×D4⋊C4).7C2, (C2×C4).1342(C2×D4), (C2×C4⋊C4).71C22, (C2×C4).580(C4○D4), (C2×C4).403(C22×C4), SmallGroup(128,666)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).57D8
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=b2c-1 >
Subgroups: 340 in 146 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×8], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, C4×C8 [×2], D4⋊C4 [×4], D4⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16 [×2], C23.65C23, C24.3C22, C2×C4×C8, C2×D4⋊C4 [×2], (C2×C4).57D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×D8, C2×SD16, C4○D8 [×2], C24.C22, C4×D8, C4×SD16, C8⋊8D4, C8⋊7D4, C4.4D8, C42.78C22, (C2×C4).57D8
►On 64 points
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 49)(16 50)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(1 18 49 31)(2 19 50 32)(3 20 51 25)(4 21 52 26)(5 22 53 27)(6 23 54 28)(7 24 55 29)(8 17 56 30)(9 47 59 39)(10 48 60 40)(11 41 61 33)(12 42 62 34)(13 43 63 35)(14 44 64 36)(15 45 57 37)(16 46 58 38)
(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 49)(2 8)(3 55)(4 6)(5 53)(7 51)(9 63)(10 12)(11 61)(13 59)(14 16)(15 57)(17 38)(18 45)(19 36)(20 43)(21 34)(22 41)(23 40)(24 47)(25 35)(26 42)(27 33)(28 48)(29 39)(30 46)(31 37)(32 44)(50 56)(52 54)(58 64)(60 62)
G:=sub<Sym(64)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (1,18,49,31)(2,19,50,32)(3,20,51,25)(4,21,52,26)(5,22,53,27)(6,23,54,28)(7,24,55,29)(8,17,56,30)(9,47,59,39)(10,48,60,40)(11,41,61,33)(12,42,62,34)(13,43,63,35)(14,44,64,36)(15,45,57,37)(16,46,58,38), (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), (1,49)(2,8)(3,55)(4,6)(5,53)(7,51)(9,63)(10,12)(11,61)(13,59)(14,16)(15,57)(17,38)(18,45)(19,36)(20,43)(21,34)(22,41)(23,40)(24,47)(25,35)(26,42)(27,33)(28,48)(29,39)(30,46)(31,37)(32,44)(50,56)(52,54)(58,64)(60,62)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (1,18,49,31)(2,19,50,32)(3,20,51,25)(4,21,52,26)(5,22,53,27)(6,23,54,28)(7,24,55,29)(8,17,56,30)(9,47,59,39)(10,48,60,40)(11,41,61,33)(12,42,62,34)(13,43,63,35)(14,44,64,36)(15,45,57,37)(16,46,58,38), (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), (1,49)(2,8)(3,55)(4,6)(5,53)(7,51)(9,63)(10,12)(11,61)(13,59)(14,16)(15,57)(17,38)(18,45)(19,36)(20,43)(21,34)(22,41)(23,40)(24,47)(25,35)(26,42)(27,33)(28,48)(29,39)(30,46)(31,37)(32,44)(50,56)(52,54)(58,64)(60,62) );
G=PermutationGroup([(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,49),(16,50),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(1,18,49,31),(2,19,50,32),(3,20,51,25),(4,21,52,26),(5,22,53,27),(6,23,54,28),(7,24,55,29),(8,17,56,30),(9,47,59,39),(10,48,60,40),(11,41,61,33),(12,42,62,34),(13,43,63,35),(14,44,64,36),(15,45,57,37),(16,46,58,38)], [(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)], [(1,49),(2,8),(3,55),(4,6),(5,53),(7,51),(9,63),(10,12),(11,61),(13,59),(14,16),(15,57),(17,38),(18,45),(19,36),(20,43),(21,34),(22,41),(23,40),(24,47),(25,35),(26,42),(27,33),(28,48),(29,39),(30,46),(31,37),(32,44),(50,56),(52,54),(58,64),(60,62)])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 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 | C4 | D4 | D4 | D8 | SD16 | C4○D4 | C4○D8 |
| kernel | (C2×C4).57D8 | C22.4Q16 | C23.65C23 | C24.3C22 | C2×C4×C8 | C2×D4⋊C4 | D4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of (C2×C4).57D8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 5 | 12 |
| 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))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[13,0,0,0,0,0,14,14,0,0,0,3,14,0,0,0,0,0,12,5,0,0,0,12,12],[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] >;
(C2×C4).57D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{57}D_8 % in TeX
G:=Group("(C2xC4).57D8"); // GroupNames label
G:=SmallGroup(128,666);
// by ID
G=gap.SmallGroup(128,666);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,58,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*c*d=b^2*c^-1>;
// generators/relations