direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C2×C8.2D4, C42.246D4, C42.371C23, C8.27(C2×D4), (C2×C8).145D4, C4⋊Q8⋊66C22, C4.7(C22×D4), C8⋊C4⋊45C22, C4.16(C4⋊1D4), (C2×C8).265C23, (C2×C4).347C24, (C2×Q16)⋊55C22, (C22×Q16)⋊18C2, (C22×C4).466D4, C23.881(C2×D4), (C2×D4).113C23, (C2×Q8).101C23, (C22×SD16).5C2, C22.52(C4⋊1D4), (C22×C8).269C22, (C2×C42).853C22, C22.607(C22×D4), (C22×C4).1562C23, (C2×SD16).114C22, C4.4D4.140C22, (C22×D4).375C22, (C22×Q8).308C22, C22.114(C8.C22), (C2×C4⋊Q8)⋊38C2, (C2×C8⋊C4)⋊9C2, (C2×C4).857(C2×D4), C2.26(C2×C4⋊1D4), C2.41(C2×C8.C22), (C2×C4.4D4).41C2, SmallGroup(128,1881)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8.2D4
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >
Subgroups: 548 in 282 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C22×D4, C22×Q8, C22×Q8, C2×C8⋊C4, C8.2D4, C2×C4.4D4, C2×C4⋊Q8, C22×SD16, C22×Q16, C2×C8.2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C8.C22, C22×D4, C8.2D4, C2×C4⋊1D4, C2×C8.C22, C2×C8.2D4
►On 64 points
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 31 57 12 5 27 61 16)(2 28 58 9 6 32 62 13)(3 25 59 14 7 29 63 10)(4 30 60 11 8 26 64 15)(17 37 53 41 21 33 49 45)(18 34 54 46 22 38 50 42)(19 39 55 43 23 35 51 47)(20 36 56 48 24 40 52 44)
(1 57)(2 60)(3 63)(4 58)(5 61)(6 64)(7 59)(8 62)(9 11)(10 14)(13 15)(17 51)(18 54)(19 49)(20 52)(21 55)(22 50)(23 53)(24 56)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)
G:=sub<Sym(64)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,31,57,12,5,27,61,16)(2,28,58,9,6,32,62,13)(3,25,59,14,7,29,63,10)(4,30,60,11,8,26,64,15)(17,37,53,41,21,33,49,45)(18,34,54,46,22,38,50,42)(19,39,55,43,23,35,51,47)(20,36,56,48,24,40,52,44), (1,57)(2,60)(3,63)(4,58)(5,61)(6,64)(7,59)(8,62)(9,11)(10,14)(13,15)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,31,57,12,5,27,61,16)(2,28,58,9,6,32,62,13)(3,25,59,14,7,29,63,10)(4,30,60,11,8,26,64,15)(17,37,53,41,21,33,49,45)(18,34,54,46,22,38,50,42)(19,39,55,43,23,35,51,47)(20,36,56,48,24,40,52,44), (1,57)(2,60)(3,63)(4,58)(5,61)(6,64)(7,59)(8,62)(9,11)(10,14)(13,15)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45)], [(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)], [(1,31,57,12,5,27,61,16),(2,28,58,9,6,32,62,13),(3,25,59,14,7,29,63,10),(4,30,60,11,8,26,64,15),(17,37,53,41,21,33,49,45),(18,34,54,46,22,38,50,42),(19,39,55,43,23,35,51,47),(20,36,56,48,24,40,52,44)], [(1,57),(2,60),(3,63),(4,58),(5,61),(6,64),(7,59),(8,62),(9,11),(10,14),(13,15),(17,51),(18,54),(19,49),(20,52),(21,55),(22,50),(23,53),(24,56),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C8.C22 |
| kernel | C2×C8.2D4 | C2×C8⋊C4 | C8.2D4 | C2×C4.4D4 | C2×C4⋊Q8 | C22×SD16 | C22×Q16 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 1 | 8 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 4 |
Matrix representation of C2×C8.2D4 ►in GL8(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 11 | 8 | 9 |
| 0 | 0 | 0 | 0 | 16 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 6 | 7 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 10 | 7 | 16 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 11 | 6 |
| 0 | 0 | 0 | 0 | 7 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 8 | 1 | 7 | 10 |
| 0 | 0 | 0 | 0 | 10 | 16 | 1 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 1 | 16 | 0 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,16,6,1,0,0,0,0,11,0,7,10,0,0,0,0,8,9,16,7,0,0,0,0,9,0,1,16],[1,15,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,7,8,10,0,0,0,0,9,0,1,16,0,0,0,0,11,6,7,1,0,0,0,0,6,0,10,7],[1,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,16,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0] >;
C2×C8.2D4 in GAP, Magma, Sage, TeX
C_2\times C_8._2D_4
% in TeX
G:=Group("C2xC8.2D4"); // GroupNames label
G:=SmallGroup(128,1881);
// by ID
G=gap.SmallGroup(128,1881);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,184,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
// generators/relations