direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C8.4Q8, C23.14Q16, C4.90(C2×D8), C8.25(C4⋊C4), (C2×C8).55Q8, C8.24(C2×Q8), (C2×C16).14C4, C16.21(C2×C4), (C2×C8).270D4, (C2×C4).171D8, C4○(C8.4Q8), (C2×C4).42Q16, C8.55(C22×C4), C4.18(C2.D8), C22.1(C2×Q16), (C22×C16).14C2, (C2×C16).94C22, (C2×C8).578C23, (C22×C4).589D4, C22.14(C2.D8), C8.C4.14C22, (C22×C8).554C22, C4.54(C2×C4⋊C4), C2.15(C2×C2.D8), (C2×C8).227(C2×C4), (C2×C4).766(C2×D4), (C2×C4).145(C4⋊C4), (C2×C8.C4).24C2, SmallGroup(128,892)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8.4Q8
G = < a,b,c,d | a2=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >
Subgroups: 108 in 72 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C8, C2×C4, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, C2×C16, C22×C8, C2×M4(2), C8.4Q8, C2×C8.C4, C22×C16, C2×C8.4Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C8.4Q8, C2×C2.D8, C2×C8.4Q8
►On 64 points
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 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 24 13 28 9 32 5 20)(2 23 14 27 10 31 6 19)(3 22 15 26 11 30 7 18)(4 21 16 25 12 29 8 17)(33 53 37 49 41 61 45 57)(34 52 38 64 42 60 46 56)(35 51 39 63 43 59 47 55)(36 50 40 62 44 58 48 54)
G:=sub<Sym(64)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,59,53,63,57,51,61,55)(50,60,54,64,58,52,62,56), (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), (1,24,13,28,9,32,5,20)(2,23,14,27,10,31,6,19)(3,22,15,26,11,30,7,18)(4,21,16,25,12,29,8,17)(33,53,37,49,41,61,45,57)(34,52,38,64,42,60,46,56)(35,51,39,63,43,59,47,55)(36,50,40,62,44,58,48,54)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,59,53,63,57,51,61,55)(50,60,54,64,58,52,62,56), (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), (1,24,13,28,9,32,5,20)(2,23,14,27,10,31,6,19)(3,22,15,26,11,30,7,18)(4,21,16,25,12,29,8,17)(33,53,37,49,41,61,45,57)(34,52,38,64,42,60,46,56)(35,51,39,63,43,59,47,55)(36,50,40,62,44,58,48,54) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,33)], [(1,11,5,15,9,3,13,7),(2,12,6,16,10,4,14,8),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,36,38,40,42,44,46,48),(49,59,53,63,57,51,61,55),(50,60,54,64,58,52,62,56)], [(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)], [(1,24,13,28,9,32,5,20),(2,23,14,27,10,31,6,19),(3,22,15,26,11,30,7,18),(4,21,16,25,12,29,8,17),(33,53,37,49,41,61,45,57),(34,52,38,64,42,60,46,56),(35,51,39,63,43,59,47,55),(36,50,40,62,44,58,48,54)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | ··· | 8P | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | Q16 | C8.4Q8 |
| kernel | C2×C8.4Q8 | C8.4Q8 | C2×C8.C4 | C22×C16 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 16 |
Matrix representation of C2×C8.4Q8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 16 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 4 | 15 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 7 | 7 |
| 13 | 0 | 0 |
| 0 | 7 | 2 |
| 0 | 3 | 10 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,9,4,0,0,15],[1,0,0,0,5,7,0,0,7],[13,0,0,0,7,3,0,2,10] >;
C2×C8.4Q8 in GAP, Magma, Sage, TeX
C_2\times C_8._4Q_8
% in TeX
G:=Group("C2xC8.4Q8"); // GroupNames label
G:=SmallGroup(128,892);
// by ID
G=gap.SmallGroup(128,892);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,1123,360,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations