p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).169D4, (C2×D4).16Q8, C42⋊9C4⋊8C2, (C2×C4).43SD16, C2.17(C8⋊3D4), C2.14(C8⋊5D4), C23.941(C2×D4), (C22×C4).325D4, C4.41(C22⋊Q8), C2.10(D4⋊2Q8), C22.80(C4⋊1D4), (C22×C8).331C22, (C2×C42).392C22, C22.108(C2×SD16), (C22×D4).97C22, C2.7(C23.4Q8), C22.159(C8⋊C22), (C22×C4).1475C23, C22.7C42⋊32C2, C4.35(C22.D4), C22.118(C22⋊Q8), C2.10(C23.46D4), C24.3C22.23C2, C22.128(C22.D4), (C2×C4.Q8)⋊26C2, (C2×C4).750(C2×D4), (C2×C4).292(C2×Q8), (C2×D4⋊C4).30C2, (C2×C4).781(C4○D4), (C2×C4⋊C4).164C22, SmallGroup(128,826)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).169D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b3, dcd-1=c3 >
Subgroups: 352 in 145 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C42⋊9C4, C24.3C22, C2×D4⋊C4, C2×C4.Q8, (C2×C8).169D4
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C2×SD16, C8⋊C22, C23.4Q8, D4⋊2Q8, C23.46D4, C8⋊5D4, C8⋊3D4, (C2×C8).169D4
►On 64 points
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(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 20 29 39 5 24 25 35)(2 54 30 47 6 50 26 43)(3 22 31 33 7 18 27 37)(4 56 32 41 8 52 28 45)(9 42 59 53 13 46 63 49)(10 36 60 21 14 40 64 17)(11 44 61 55 15 48 57 51)(12 38 62 23 16 34 58 19)
(1 40 59 47)(2 35 60 42)(3 38 61 45)(4 33 62 48)(5 36 63 43)(6 39 64 46)(7 34 57 41)(8 37 58 44)(9 50 25 17)(10 53 26 20)(11 56 27 23)(12 51 28 18)(13 54 29 21)(14 49 30 24)(15 52 31 19)(16 55 32 22)
G:=sub<Sym(64)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,20,29,39,5,24,25,35)(2,54,30,47,6,50,26,43)(3,22,31,33,7,18,27,37)(4,56,32,41,8,52,28,45)(9,42,59,53,13,46,63,49)(10,36,60,21,14,40,64,17)(11,44,61,55,15,48,57,51)(12,38,62,23,16,34,58,19), (1,40,59,47)(2,35,60,42)(3,38,61,45)(4,33,62,48)(5,36,63,43)(6,39,64,46)(7,34,57,41)(8,37,58,44)(9,50,25,17)(10,53,26,20)(11,56,27,23)(12,51,28,18)(13,54,29,21)(14,49,30,24)(15,52,31,19)(16,55,32,22)>;
G:=Group( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,20,29,39,5,24,25,35)(2,54,30,47,6,50,26,43)(3,22,31,33,7,18,27,37)(4,56,32,41,8,52,28,45)(9,42,59,53,13,46,63,49)(10,36,60,21,14,40,64,17)(11,44,61,55,15,48,57,51)(12,38,62,23,16,34,58,19), (1,40,59,47)(2,35,60,42)(3,38,61,45)(4,33,62,48)(5,36,63,43)(6,39,64,46)(7,34,57,41)(8,37,58,44)(9,50,25,17)(10,53,26,20)(11,56,27,23)(12,51,28,18)(13,54,29,21)(14,49,30,24)(15,52,31,19)(16,55,32,22) );
G=PermutationGroup([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(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)], [(1,20,29,39,5,24,25,35),(2,54,30,47,6,50,26,43),(3,22,31,33,7,18,27,37),(4,56,32,41,8,52,28,45),(9,42,59,53,13,46,63,49),(10,36,60,21,14,40,64,17),(11,44,61,55,15,48,57,51),(12,38,62,23,16,34,58,19)], [(1,40,59,47),(2,35,60,42),(3,38,61,45),(4,33,62,48),(5,36,63,43),(6,39,64,46),(7,34,57,41),(8,37,58,44),(9,50,25,17),(10,53,26,20),(11,56,27,23),(12,51,28,18),(13,54,29,21),(14,49,30,24),(15,52,31,19),(16,55,32,22)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | SD16 | C4○D4 | C8⋊C22 |
| kernel | (C2×C8).169D4 | C22.7C42 | C42⋊9C4 | C24.3C22 | C2×D4⋊C4 | C2×C4.Q8 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 6 | 2 |
Matrix representation of (C2×C8).169D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,2,0,0,0,0,9,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,0,9,0,0,0,0,2,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;
(C2×C8).169D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{169}D_4 % in TeX
G:=Group("(C2xC8).169D4"); // GroupNames label
G:=SmallGroup(128,826);
// by ID
G=gap.SmallGroup(128,826);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,624,422,387,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations