p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4○D4⋊4D4, (C2×Q8)⋊17D4, Q8⋊D4⋊4C2, D4.48(C2×D4), Q8.48(C2×D4), C2.7(Q8○D8), D4⋊D4⋊18C2, C4.85C22≀C2, (C2×D4).296D4, C4⋊C4.15C23, C4.50(C22×D4), D4.7D4⋊18C2, (C2×C8).131C23, (C2×C4).232C24, C22⋊Q16⋊14C2, C22.4C22≀C2, (C2×D4).32C23, C23.236(C2×D4), C2.11(D4○SD16), (C2×D8).115C22, C4⋊D4.18C22, C23.36D4⋊4C2, C22⋊C8.13C22, (C2×2- 1+4)⋊2C2, (C2×Q8).356C23, C22⋊Q8.18C22, D4⋊C4.19C22, (C22×C8).176C22, (C22×C4).280C23, Q8⋊C4.21C22, (C2×Q16).114C22, C22.492(C22×D4), C22.31C24⋊4C2, (C2×SD16).130C22, (C2×M4(2)).43C22, (C22×Q8).263C22, (C2×C4○D8)⋊4C2, (C2×C4).459(C2×D4), (C22×C8)⋊C2⋊8C2, (C2×Q8⋊C4)⋊28C2, C2.50(C2×C22≀C2), (C2×C8.C22)⋊10C2, (C2×C4⋊C4).582C22, (C2×C4○D4).103C22, SmallGroup(128,1745)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×Q8)⋊17D4
G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=b-1, dbd-1=ab-1c, ebe=abc, dcd-1=b2c, ce=ec, ede=d-1 >
Subgroups: 660 in 356 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C8.C22, C22×Q8, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, (C22×C8)⋊C2, C2×Q8⋊C4, C23.36D4, Q8⋊D4, D4⋊D4, C22⋊Q16, D4.7D4, C22.31C24, C2×C4○D8, C2×C8.C22, C2×2- 1+4, (C2×Q8)⋊17D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, D4○SD16, Q8○D8, (C2×Q8)⋊17D4
►On 64 points
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 45)(10 46)(11 47)(12 48)(13 52)(14 49)(15 50)(16 51)(17 37)(18 38)(19 39)(20 40)(21 44)(22 41)(23 42)(24 43)(25 64)(26 61)(27 62)(28 63)(29 33)(30 34)(31 35)(32 36)
(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 36 3 34)(2 35 4 33)(5 62 7 64)(6 61 8 63)(9 42 11 44)(10 41 12 43)(13 37 15 39)(14 40 16 38)(17 50 19 52)(18 49 20 51)(21 45 23 47)(22 48 24 46)(25 57 27 59)(26 60 28 58)(29 54 31 56)(30 53 32 55)
(1 9 62 38)(2 24 63 50)(3 11 64 40)(4 22 61 52)(5 14 34 42)(6 17 35 48)(7 16 36 44)(8 19 33 46)(10 60 39 29)(12 58 37 31)(13 56 41 26)(15 54 43 28)(18 53 45 27)(20 55 47 25)(21 59 51 32)(23 57 49 30)
(2 31)(4 29)(6 26)(8 28)(9 38)(10 52)(11 40)(12 50)(13 48)(14 42)(15 46)(16 44)(17 41)(18 47)(19 43)(20 45)(21 49)(22 39)(23 51)(24 37)(25 27)(30 32)(33 54)(35 56)(53 55)(57 59)(58 63)(60 61)
G:=sub<Sym(64)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,45)(10,46)(11,47)(12,48)(13,52)(14,49)(15,50)(16,51)(17,37)(18,38)(19,39)(20,40)(21,44)(22,41)(23,42)(24,43)(25,64)(26,61)(27,62)(28,63)(29,33)(30,34)(31,35)(32,36), (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), (1,36,3,34)(2,35,4,33)(5,62,7,64)(6,61,8,63)(9,42,11,44)(10,41,12,43)(13,37,15,39)(14,40,16,38)(17,50,19,52)(18,49,20,51)(21,45,23,47)(22,48,24,46)(25,57,27,59)(26,60,28,58)(29,54,31,56)(30,53,32,55), (1,9,62,38)(2,24,63,50)(3,11,64,40)(4,22,61,52)(5,14,34,42)(6,17,35,48)(7,16,36,44)(8,19,33,46)(10,60,39,29)(12,58,37,31)(13,56,41,26)(15,54,43,28)(18,53,45,27)(20,55,47,25)(21,59,51,32)(23,57,49,30), (2,31)(4,29)(6,26)(8,28)(9,38)(10,52)(11,40)(12,50)(13,48)(14,42)(15,46)(16,44)(17,41)(18,47)(19,43)(20,45)(21,49)(22,39)(23,51)(24,37)(25,27)(30,32)(33,54)(35,56)(53,55)(57,59)(58,63)(60,61)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,45)(10,46)(11,47)(12,48)(13,52)(14,49)(15,50)(16,51)(17,37)(18,38)(19,39)(20,40)(21,44)(22,41)(23,42)(24,43)(25,64)(26,61)(27,62)(28,63)(29,33)(30,34)(31,35)(32,36), (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), (1,36,3,34)(2,35,4,33)(5,62,7,64)(6,61,8,63)(9,42,11,44)(10,41,12,43)(13,37,15,39)(14,40,16,38)(17,50,19,52)(18,49,20,51)(21,45,23,47)(22,48,24,46)(25,57,27,59)(26,60,28,58)(29,54,31,56)(30,53,32,55), (1,9,62,38)(2,24,63,50)(3,11,64,40)(4,22,61,52)(5,14,34,42)(6,17,35,48)(7,16,36,44)(8,19,33,46)(10,60,39,29)(12,58,37,31)(13,56,41,26)(15,54,43,28)(18,53,45,27)(20,55,47,25)(21,59,51,32)(23,57,49,30), (2,31)(4,29)(6,26)(8,28)(9,38)(10,52)(11,40)(12,50)(13,48)(14,42)(15,46)(16,44)(17,41)(18,47)(19,43)(20,45)(21,49)(22,39)(23,51)(24,37)(25,27)(30,32)(33,54)(35,56)(53,55)(57,59)(58,63)(60,61) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,45),(10,46),(11,47),(12,48),(13,52),(14,49),(15,50),(16,51),(17,37),(18,38),(19,39),(20,40),(21,44),(22,41),(23,42),(24,43),(25,64),(26,61),(27,62),(28,63),(29,33),(30,34),(31,35),(32,36)], [(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)], [(1,36,3,34),(2,35,4,33),(5,62,7,64),(6,61,8,63),(9,42,11,44),(10,41,12,43),(13,37,15,39),(14,40,16,38),(17,50,19,52),(18,49,20,51),(21,45,23,47),(22,48,24,46),(25,57,27,59),(26,60,28,58),(29,54,31,56),(30,53,32,55)], [(1,9,62,38),(2,24,63,50),(3,11,64,40),(4,22,61,52),(5,14,34,42),(6,17,35,48),(7,16,36,44),(8,19,33,46),(10,60,39,29),(12,58,37,31),(13,56,41,26),(15,54,43,28),(18,53,45,27),(20,55,47,25),(21,59,51,32),(23,57,49,30)], [(2,31),(4,29),(6,26),(8,28),(9,38),(10,52),(11,40),(12,50),(13,48),(14,42),(15,46),(16,44),(17,41),(18,47),(19,43),(20,45),(21,49),(22,39),(23,51),(24,37),(25,27),(30,32),(33,54),(35,56),(53,55),(57,59),(58,63),(60,61)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4○SD16 | Q8○D8 |
| kernel | (C2×Q8)⋊17D4 | (C22×C8)⋊C2 | C2×Q8⋊C4 | C23.36D4 | Q8⋊D4 | D4⋊D4 | C22⋊Q16 | D4.7D4 | C22.31C24 | C2×C4○D8 | C2×C8.C22 | C2×2- 1+4 | C2×D4 | C2×Q8 | C4○D4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 5 | 4 | 2 | 2 |
Matrix representation of (C2×Q8)⋊17D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 4 | 4 |
| 0 | 0 | 4 | 4 | 0 | 4 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 12 | 7 |
| 0 | 0 | 5 | 0 | 5 | 0 |
| 0 | 0 | 5 | 10 | 5 | 10 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 | 16 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,16,1,0,0,16,16,0,1,0,0,15,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13,4,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,1,16,0,0,0,1,0,16,0,0,16,1,0,0,0,0,0,2,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,12,5,5,12,0,0,0,0,10,12,0,0,12,5,5,0,0,0,7,0,10,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
(C2×Q8)⋊17D4 in GAP, Magma, Sage, TeX
(C_2\times Q_8)\rtimes_{17}D_4 % in TeX
G:=Group("(C2xQ8):17D4"); // GroupNames label
G:=SmallGroup(128,1745);
// by ID
G=gap.SmallGroup(128,1745);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,248,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1*c,e*b*e=a*b*c,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations