direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊8D4, C23⋊5SD16, C24.141D4, C8⋊17(C2×D4), (C2×C8)⋊39D4, (C23×C8)⋊12C2, C4⋊C4.16C23, C22⋊1(C2×SD16), C4.Q8⋊54C22, (C2×C8).485C23, (C2×C4).251C24, (C22×C8)⋊68C22, (C2×D4).56C23, C4.145(C22×D4), C23.858(C2×D4), (C22×C4).605D4, C22⋊Q8⋊65C22, (C2×Q8).44C23, C4.107(C4⋊D4), D4⋊C4⋊56C22, C2.9(C22×SD16), Q8⋊C4⋊57C22, (C2×SD16)⋊75C22, (C22×SD16)⋊25C2, C22.90(C4○D8), C4⋊D4.146C22, (C23×C4).700C22, C22.511(C22×D4), C22.171(C4⋊D4), (C22×C4).1530C23, (C22×D4).344C22, (C22×Q8).277C22, (C2×C4.Q8)⋊27C2, C4.18(C2×C4○D4), C2.13(C2×C4○D8), C2.69(C2×C4⋊D4), (C2×C22⋊Q8)⋊54C2, (C2×D4⋊C4)⋊16C2, (C2×Q8⋊C4)⋊17C2, (C2×C4).1423(C2×D4), (C2×C4⋊D4).53C2, (C2×C4).697(C4○D4), (C2×C4⋊C4).585C22, SmallGroup(128,1779)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊8D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >
Subgroups: 564 in 282 conjugacy classes, 116 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, D4⋊C4, Q8⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C22×C8, C22×C8, C2×SD16, C2×SD16, C23×C4, C22×D4, C22×D4, C22×Q8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C8⋊8D4, C2×C4⋊D4, C2×C22⋊Q8, C23×C8, C22×SD16, C2×C8⋊8D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C4⋊D4, C2×SD16, C4○D8, C22×D4, C2×C4○D4, C8⋊8D4, C2×C4⋊D4, C22×SD16, C2×C4○D8, C2×C8⋊8D4
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(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 37 23 45)(2 40 24 48)(3 35 17 43)(4 38 18 46)(5 33 19 41)(6 36 20 44)(7 39 21 47)(8 34 22 42)(9 55 63 25)(10 50 64 28)(11 53 57 31)(12 56 58 26)(13 51 59 29)(14 54 60 32)(15 49 61 27)(16 52 62 30)
(1 25)(2 28)(3 31)(4 26)(5 29)(6 32)(7 27)(8 30)(9 45)(10 48)(11 43)(12 46)(13 41)(14 44)(15 47)(16 42)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(33 59)(34 62)(35 57)(36 60)(37 63)(38 58)(39 61)(40 64)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (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), (1,37,23,45)(2,40,24,48)(3,35,17,43)(4,38,18,46)(5,33,19,41)(6,36,20,44)(7,39,21,47)(8,34,22,42)(9,55,63,25)(10,50,64,28)(11,53,57,31)(12,56,58,26)(13,51,59,29)(14,54,60,32)(15,49,61,27)(16,52,62,30), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,45)(10,48)(11,43)(12,46)(13,41)(14,44)(15,47)(16,42)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(33,59)(34,62)(35,57)(36,60)(37,63)(38,58)(39,61)(40,64)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (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), (1,37,23,45)(2,40,24,48)(3,35,17,43)(4,38,18,46)(5,33,19,41)(6,36,20,44)(7,39,21,47)(8,34,22,42)(9,55,63,25)(10,50,64,28)(11,53,57,31)(12,56,58,26)(13,51,59,29)(14,54,60,32)(15,49,61,27)(16,52,62,30), (1,25)(2,28)(3,31)(4,26)(5,29)(6,32)(7,27)(8,30)(9,45)(10,48)(11,43)(12,46)(13,41)(14,44)(15,47)(16,42)(17,53)(18,56)(19,51)(20,54)(21,49)(22,52)(23,55)(24,50)(33,59)(34,62)(35,57)(36,60)(37,63)(38,58)(39,61)(40,64) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(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)], [(1,37,23,45),(2,40,24,48),(3,35,17,43),(4,38,18,46),(5,33,19,41),(6,36,20,44),(7,39,21,47),(8,34,22,42),(9,55,63,25),(10,50,64,28),(11,53,57,31),(12,56,58,26),(13,51,59,29),(14,54,60,32),(15,49,61,27),(16,52,62,30)], [(1,25),(2,28),(3,31),(4,26),(5,29),(6,32),(7,27),(8,30),(9,45),(10,48),(11,43),(12,46),(13,41),(14,44),(15,47),(16,42),(17,53),(18,56),(19,51),(20,54),(21,49),(22,52),(23,55),(24,50),(33,59),(34,62),(35,57),(36,60),(37,63),(38,58),(39,61),(40,64)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | SD16 | C4○D8 |
| kernel | C2×C8⋊8D4 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.Q8 | C8⋊8D4 | C2×C4⋊D4 | C2×C22⋊Q8 | C23×C8 | C22×SD16 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 8 | 8 |
Matrix representation of C2×C8⋊8D4 ►in 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 | 12 | 5 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 5 | 5 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 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))| [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],[1,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,12,5],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,1],[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×C8⋊8D4 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_8D_4
% in TeX
G:=Group("C2xC8:8D4"); // GroupNames label
G:=SmallGroup(128,1779);
// by ID
G=gap.SmallGroup(128,1779);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172]);
// 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^3,d*c*d=c^-1>;
// generators/relations