p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.418D4, C42.172C23, (C4×Q8).8C4, C22⋊Q8.15C4, C4.102(C4○D8), C4.10D8⋊38C2, C4⋊C8.262C22, C42.113(C2×C4), C4.6Q16⋊23C2, (C22×C4).241D4, C4⋊Q8.245C22, C4.106(C8.C22), C23.65(C22⋊C4), C42.6C4.24C2, (C2×C42).216C22, C42.12C4.27C2, C2.13(C23.38D4), C2.19(C23.24D4), C23.37C23.16C2, C2.19(M4(2).8C22), C4⋊C4.42(C2×C4), (C2×Q8).33(C2×C4), (C2×C4).1243(C2×D4), (C2×C4).166(C22×C4), (C22×C4).238(C2×C4), (C2×C4).187(C22⋊C4), C22.230(C2×C22⋊C4), SmallGroup(128,286)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.418D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=bc3 >
Subgroups: 180 in 99 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4.10D8, C4.6Q16, C42.12C4, C42.6C4, C23.37C23, C42.418D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C4○D8, C8.C22, M4(2).8C22, C23.24D4, C23.38D4, C42.418D4
►On 64 points
(1 3 5 7)(2 21 6 17)(4 23 8 19)(9 11 13 15)(10 58 14 62)(12 60 16 64)(18 20 22 24)(25 54 29 50)(26 32 30 28)(27 56 31 52)(33 47 37 43)(34 40 38 36)(35 41 39 45)(42 48 46 44)(49 55 53 51)(57 59 61 63)
(1 57 22 15)(2 16 23 58)(3 59 24 9)(4 10 17 60)(5 61 18 11)(6 12 19 62)(7 63 20 13)(8 14 21 64)(25 43 52 39)(26 40 53 44)(27 45 54 33)(28 34 55 46)(29 47 56 35)(30 36 49 48)(31 41 50 37)(32 38 51 42)
(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 48 57 30 22 36 15 49)(2 52 16 39 23 25 58 43)(3 46 59 28 24 34 9 55)(4 50 10 37 17 31 60 41)(5 44 61 26 18 40 11 53)(6 56 12 35 19 29 62 47)(7 42 63 32 20 38 13 51)(8 54 14 33 21 27 64 45)
G:=sub<Sym(64)| (1,3,5,7)(2,21,6,17)(4,23,8,19)(9,11,13,15)(10,58,14,62)(12,60,16,64)(18,20,22,24)(25,54,29,50)(26,32,30,28)(27,56,31,52)(33,47,37,43)(34,40,38,36)(35,41,39,45)(42,48,46,44)(49,55,53,51)(57,59,61,63), (1,57,22,15)(2,16,23,58)(3,59,24,9)(4,10,17,60)(5,61,18,11)(6,12,19,62)(7,63,20,13)(8,14,21,64)(25,43,52,39)(26,40,53,44)(27,45,54,33)(28,34,55,46)(29,47,56,35)(30,36,49,48)(31,41,50,37)(32,38,51,42), (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), (1,48,57,30,22,36,15,49)(2,52,16,39,23,25,58,43)(3,46,59,28,24,34,9,55)(4,50,10,37,17,31,60,41)(5,44,61,26,18,40,11,53)(6,56,12,35,19,29,62,47)(7,42,63,32,20,38,13,51)(8,54,14,33,21,27,64,45)>;
G:=Group( (1,3,5,7)(2,21,6,17)(4,23,8,19)(9,11,13,15)(10,58,14,62)(12,60,16,64)(18,20,22,24)(25,54,29,50)(26,32,30,28)(27,56,31,52)(33,47,37,43)(34,40,38,36)(35,41,39,45)(42,48,46,44)(49,55,53,51)(57,59,61,63), (1,57,22,15)(2,16,23,58)(3,59,24,9)(4,10,17,60)(5,61,18,11)(6,12,19,62)(7,63,20,13)(8,14,21,64)(25,43,52,39)(26,40,53,44)(27,45,54,33)(28,34,55,46)(29,47,56,35)(30,36,49,48)(31,41,50,37)(32,38,51,42), (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), (1,48,57,30,22,36,15,49)(2,52,16,39,23,25,58,43)(3,46,59,28,24,34,9,55)(4,50,10,37,17,31,60,41)(5,44,61,26,18,40,11,53)(6,56,12,35,19,29,62,47)(7,42,63,32,20,38,13,51)(8,54,14,33,21,27,64,45) );
G=PermutationGroup([[(1,3,5,7),(2,21,6,17),(4,23,8,19),(9,11,13,15),(10,58,14,62),(12,60,16,64),(18,20,22,24),(25,54,29,50),(26,32,30,28),(27,56,31,52),(33,47,37,43),(34,40,38,36),(35,41,39,45),(42,48,46,44),(49,55,53,51),(57,59,61,63)], [(1,57,22,15),(2,16,23,58),(3,59,24,9),(4,10,17,60),(5,61,18,11),(6,12,19,62),(7,63,20,13),(8,14,21,64),(25,43,52,39),(26,40,53,44),(27,45,54,33),(28,34,55,46),(29,47,56,35),(30,36,49,48),(31,41,50,37),(32,38,51,42)], [(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)], [(1,48,57,30,22,36,15,49),(2,52,16,39,23,25,58,43),(3,46,59,28,24,34,9,55),(4,50,10,37,17,31,60,41),(5,44,61,26,18,40,11,53),(6,56,12,35,19,29,62,47),(7,42,63,32,20,38,13,51),(8,54,14,33,21,27,64,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D8 | C8.C22 | M4(2).8C22 |
| kernel | C42.418D4 | C4.10D8 | C4.6Q16 | C42.12C4 | C42.6C4 | C23.37C23 | C4×Q8 | C22⋊Q8 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.418D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 4 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 6 |
| 0 | 0 | 4 | 0 | 16 | 10 |
| 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 4 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 1 | 11 |
| 0 | 0 | 0 | 4 | 0 | 6 |
| 0 | 0 | 4 | 13 | 0 | 4 |
| 0 | 0 | 0 | 8 | 0 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,4,0,4,13],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,13,4,13,8,0,0,0,0,13,0,0,0,0,16,0,0,0,0,6,10,0,4],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,4,0,0,0,13,4,13,8,0,0,1,0,0,0,0,0,11,6,4,13] >;
C42.418D4 in GAP, Magma, Sage, TeX
C_4^2._{418}D_4 % in TeX
G:=Group("C4^2.418D4"); // GroupNames label
G:=SmallGroup(128,286);
// by ID
G=gap.SmallGroup(128,286);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,520,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations