p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.21D8, C16.12D4, C42.339D4, (C4×C16)⋊11C2, (C2×Q32)⋊6C2, C8.42(C2×D4), (C2×C4).63D8, C4.10(C2×D8), (C2×D16).3C2, (C2×C8).280D4, (C2×SD32)⋊17C2, C8.12D4⋊2C2, C4.4(C4⋊1D4), C2.18(C4○D16), C2.15(C8⋊4D4), (C4×C8).414C22, (C2×C8).547C23, (C2×C16).85C22, (C2×D8).18C22, C22.133(C2×D8), (C2×Q16).19C22, (C2×C4).815(C2×D4), SmallGroup(128,981)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.21D8
G = < a,b,c | a8=c2=1, b8=a4, ab=ba, cac=a3, cbc=a4b7 >
Subgroups: 264 in 92 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C2×C16, D16, SD32, Q32, C4.4D4, C2×D8, C2×SD16, C2×Q16, C4×C16, C8.12D4, C2×D16, C2×SD32, C2×Q32, C8.21D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C8⋊4D4, C4○D16, C8.21D8
►On 64 points
(1 51 30 45 9 59 22 37)(2 52 31 46 10 60 23 38)(3 53 32 47 11 61 24 39)(4 54 17 48 12 62 25 40)(5 55 18 33 13 63 26 41)(6 56 19 34 14 64 27 42)(7 57 20 35 15 49 28 43)(8 58 21 36 16 50 29 44)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 64)(48 63)
G:=sub<Sym(64)| (1,51,30,45,9,59,22,37)(2,52,31,46,10,60,23,38)(3,53,32,47,11,61,24,39)(4,54,17,48,12,62,25,40)(5,55,18,33,13,63,26,41)(6,56,19,34,14,64,27,42)(7,57,20,35,15,49,28,43)(8,58,21,36,16,50,29,44), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,64)(48,63)>;
G:=Group( (1,51,30,45,9,59,22,37)(2,52,31,46,10,60,23,38)(3,53,32,47,11,61,24,39)(4,54,17,48,12,62,25,40)(5,55,18,33,13,63,26,41)(6,56,19,34,14,64,27,42)(7,57,20,35,15,49,28,43)(8,58,21,36,16,50,29,44), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,64)(48,63) );
G=PermutationGroup([[(1,51,30,45,9,59,22,37),(2,52,31,46,10,60,23,38),(3,53,32,47,11,61,24,39),(4,54,17,48,12,62,25,40),(5,55,18,33,13,63,26,41),(6,56,19,34,14,64,27,42),(7,57,20,35,15,49,28,43),(8,58,21,36,16,50,29,44)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(33,62),(34,61),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,64),(48,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | ··· | 2 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | D8 | C4○D16 |
| kernel | C8.21D8 | C4×C16 | C8.12D4 | C2×D16 | C2×SD32 | C2×Q32 | C16 | C42 | C2×C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C8.21D8 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 5 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 7 | 12 |
| 0 | 0 | 11 | 2 |
| 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 7 | 12 |
| 0 | 0 | 13 | 10 |
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,10,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,7,11,0,0,12,2],[16,1,0,0,0,1,0,0,0,0,7,13,0,0,12,10] >;
C8.21D8 in GAP, Magma, Sage, TeX
C_8._{21}D_8 % in TeX
G:=Group("C8.21D8"); // GroupNames label
G:=SmallGroup(128,981);
// by ID
G=gap.SmallGroup(128,981);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,436,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,c*a*c=a^3,c*b*c=a^4*b^7>;
// generators/relations