p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.79D4, C42.168C23, (C4×Q8).7C4, C4.38(C2×Q16), (C2×C4).14Q16, C4.56(C2×SD16), (C2×C4).31SD16, C22⋊Q8.14C4, C4.10D8⋊35C2, C4⋊C8.260C22, C4.83(C8⋊C22), C42.109(C2×C4), C4.6Q16⋊21C2, (C22×C4).744D4, C4⋊Q8.241C22, C4.26(Q8⋊C4), C4.85(C8.C22), C4⋊M4(2).17C2, (C2×C42).212C22, C23.110(C22⋊C4), C22.12(Q8⋊C4), C2.13(C23.36D4), C23.37C23.14C2, C2.17(M4(2).8C22), (C2×C4⋊C8).14C2, C4⋊C4.39(C2×C4), (C2×Q8).31(C2×C4), (C2×C4).1239(C2×D4), C2.16(C2×Q8⋊C4), (C22×C4).234(C2×C4), (C2×C4).162(C22×C4), (C2×C4).183(C22⋊C4), C22.226(C2×C22⋊C4), SmallGroup(128,282)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.79D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 196 in 108 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C4.10D8, C4.6Q16, C2×C4⋊C8, C4⋊M4(2), C23.37C23, C42.79D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C8⋊C22, C8.C22, M4(2).8C22, C2×Q8⋊C4, C23.36D4, C42.79D4
►On 64 points
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(1 16 58 22)(2 23 59 9)(3 10 60 24)(4 17 61 11)(5 12 62 18)(6 19 63 13)(7 14 64 20)(8 21 57 15)(25 38 50 41)(26 42 51 39)(27 40 52 43)(28 44 53 33)(29 34 54 45)(30 46 55 35)(31 36 56 47)(32 48 49 37)
(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 16 49 58 37 22 32)(2 52 23 43 59 27 9 40)(3 46 10 55 60 35 24 30)(4 50 17 41 61 25 11 38)(5 44 12 53 62 33 18 28)(6 56 19 47 63 31 13 36)(7 42 14 51 64 39 20 26)(8 54 21 45 57 29 15 34)
G:=sub<Sym(64)| (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (1,16,58,22)(2,23,59,9)(3,10,60,24)(4,17,61,11)(5,12,62,18)(6,19,63,13)(7,14,64,20)(8,21,57,15)(25,38,50,41)(26,42,51,39)(27,40,52,43)(28,44,53,33)(29,34,54,45)(30,46,55,35)(31,36,56,47)(32,48,49,37), (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), (1,48,16,49,58,37,22,32)(2,52,23,43,59,27,9,40)(3,46,10,55,60,35,24,30)(4,50,17,41,61,25,11,38)(5,44,12,53,62,33,18,28)(6,56,19,47,63,31,13,36)(7,42,14,51,64,39,20,26)(8,54,21,45,57,29,15,34)>;
G:=Group( (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (1,16,58,22)(2,23,59,9)(3,10,60,24)(4,17,61,11)(5,12,62,18)(6,19,63,13)(7,14,64,20)(8,21,57,15)(25,38,50,41)(26,42,51,39)(27,40,52,43)(28,44,53,33)(29,34,54,45)(30,46,55,35)(31,36,56,47)(32,48,49,37), (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), (1,48,16,49,58,37,22,32)(2,52,23,43,59,27,9,40)(3,46,10,55,60,35,24,30)(4,50,17,41,61,25,11,38)(5,44,12,53,62,33,18,28)(6,56,19,47,63,31,13,36)(7,42,14,51,64,39,20,26)(8,54,21,45,57,29,15,34) );
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,39,37,35),(34,36,38,40),(41,43,45,47),(42,48,46,44),(49,55,53,51),(50,52,54,56),(57,59,61,63),(58,64,62,60)], [(1,16,58,22),(2,23,59,9),(3,10,60,24),(4,17,61,11),(5,12,62,18),(6,19,63,13),(7,14,64,20),(8,21,57,15),(25,38,50,41),(26,42,51,39),(27,40,52,43),(28,44,53,33),(29,34,54,45),(30,46,55,35),(31,36,56,47),(32,48,49,37)], [(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)], [(1,48,16,49,58,37,22,32),(2,52,23,43,59,27,9,40),(3,46,10,55,60,35,24,30),(4,50,17,41,61,25,11,38),(5,44,12,53,62,33,18,28),(6,56,19,47,63,31,13,36),(7,42,14,51,64,39,20,26),(8,54,21,45,57,29,15,34)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 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 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C8⋊C22 | C8.C22 | M4(2).8C22 |
| kernel | C42.79D4 | C4.10D8 | C4.6Q16 | C2×C4⋊C8 | C4⋊M4(2) | C23.37C23 | C4×Q8 | C22⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of C42.79D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 8 | 0 |
| 0 | 0 | 13 | 0 | 4 | 4 |
| 0 | 0 | 0 | 13 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 13 | 13 | 0 |
| 0 | 0 | 0 | 13 | 0 | 4 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,13,13,0,0,0,0,0,0,13,0,0,0,8,4,4,4,0,0,0,4,0,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,4,0,4,0,0,0,9,13,13,13,0,0,0,0,13,0,0,0,0,0,0,4] >;
C42.79D4 in GAP, Magma, Sage, TeX
C_4^2._{79}D_4 % in TeX
G:=Group("C4^2.79D4"); // GroupNames label
G:=SmallGroup(128,282);
// by ID
G=gap.SmallGroup(128,282);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,758,184,1123,1018,248,1971,242]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations