direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊7D4, C23⋊4D8, C24.142D4, C8⋊16(C2×D4), (C2×C8)⋊37D4, (C23×C8)⋊8C2, C22⋊1(C2×D8), C2.6(C22×D8), (C22×D8)⋊10C2, (C2×D8)⋊42C22, C4⋊C4.17C23, C2.D8⋊44C22, C4⋊D4⋊53C22, (C2×C8).486C23, (C2×C4).252C24, (C22×C8)⋊65C22, (C2×D4).57C23, C23.859(C2×D4), (C22×C4).606D4, C4.146(C22×D4), C4.108(C4⋊D4), D4⋊C4⋊57C22, C22.91(C4○D8), (C23×C4).701C22, C22.512(C22×D4), C22.172(C4⋊D4), (C22×C4).1531C23, (C22×D4).345C22, (C2×C2.D8)⋊18C2, C2.14(C2×C4○D8), C4.19(C2×C4○D4), (C2×C4⋊D4)⋊46C2, C2.70(C2×C4⋊D4), (C2×D4⋊C4)⋊17C2, (C2×C4).1424(C2×D4), (C2×C4).698(C4○D4), (C2×C4⋊C4).586C22, SmallGroup(128,1780)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊7D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 692 in 308 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D4⋊C4, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C22×C8, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C2×D4⋊C4, C2×C2.D8, C8⋊7D4, C2×C4⋊D4, C23×C8, C22×D8, C2×C8⋊7D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C4⋊D4, C2×D8, C4○D8, C22×D4, C2×C4○D4, C8⋊7D4, C2×C4⋊D4, C22×D8, C2×C4○D8, C2×C8⋊7D4
►On 64 points
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 42 39 23)(2 41 40 22)(3 48 33 21)(4 47 34 20)(5 46 35 19)(6 45 36 18)(7 44 37 17)(8 43 38 24)(9 31 52 58)(10 30 53 57)(11 29 54 64)(12 28 55 63)(13 27 56 62)(14 26 49 61)(15 25 50 60)(16 32 51 59)
(1 51)(2 50)(3 49)(4 56)(5 55)(6 54)(7 53)(8 52)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 57)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)
G:=sub<Sym(64)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,42,39,23)(2,41,40,22)(3,48,33,21)(4,47,34,20)(5,46,35,19)(6,45,36,18)(7,44,37,17)(8,43,38,24)(9,31,52,58)(10,30,53,57)(11,29,54,64)(12,28,55,63)(13,27,56,62)(14,26,49,61)(15,25,50,60)(16,32,51,59), (1,51)(2,50)(3,49)(4,56)(5,55)(6,54)(7,53)(8,52)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,42,39,23)(2,41,40,22)(3,48,33,21)(4,47,34,20)(5,46,35,19)(6,45,36,18)(7,44,37,17)(8,43,38,24)(9,31,52,58)(10,30,53,57)(11,29,54,64)(12,28,55,63)(13,27,56,62)(14,26,49,61)(15,25,50,60)(16,32,51,59), (1,51)(2,50)(3,49)(4,56)(5,55)(6,54)(7,53)(8,52)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42) );
G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(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)], [(1,42,39,23),(2,41,40,22),(3,48,33,21),(4,47,34,20),(5,46,35,19),(6,45,36,18),(7,44,37,17),(8,43,38,24),(9,31,52,58),(10,30,53,57),(11,29,54,64),(12,28,55,63),(13,27,56,62),(14,26,49,61),(15,25,50,60),(16,32,51,59)], [(1,51),(2,50),(3,49),(4,56),(5,55),(6,54),(7,53),(8,52),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(17,57),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,41),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 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 | D8 | C4○D8 |
| kernel | C2×C8⋊7D4 | C2×D4⋊C4 | C2×C2.D8 | C8⋊7D4 | C2×C4⋊D4 | C23×C8 | C22×D8 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 8 | 8 |
Matrix representation of C2×C8⋊7D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
C2×C8⋊7D4 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_7D_4
% in TeX
G:=Group("C2xC8:7D4"); // GroupNames label
G:=SmallGroup(128,1780);
// by ID
G=gap.SmallGroup(128,1780);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations