p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.408D4, C42.146C23, C22.26C4≀C2, C42.87(C2×C4), (C22×C4).226D4, C42.C2.10C4, C4.5(C4.10D4), (C4×M4(2)).19C2, C8⋊C4.145C22, C42.6C4.19C2, (C2×C42).190C22, C42.2C22⋊6C2, C42.C2.94C22, C23.178(C22⋊C4), C2.28(C42⋊C22), C2.33(C2×C4≀C2), (C2×C4⋊C4).17C4, C4⋊C4.25(C2×C4), (C2×C4).1174(C2×D4), (C2×C42.C2).3C2, (C22×C4).212(C2×C4), (C2×C4).140(C22×C4), C2.11(C2×C4.10D4), (C2×C4).178(C22⋊C4), C22.204(C2×C22⋊C4), SmallGroup(128,260)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.408D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 188 in 106 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C2×M4(2), C42.2C22, C4×M4(2), C42.6C4, C2×C42.C2, C42.408D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C4≀C2, C2×C22⋊C4, C2×C4.10D4, C2×C4≀C2, C42⋊C22, C42.408D4
►On 64 points
(1 16 60 22)(2 23 61 9)(3 10 62 24)(4 17 63 11)(5 12 64 18)(6 19 57 13)(7 14 58 20)(8 21 59 15)(25 38 52 48)(26 41 53 39)(27 40 54 42)(28 43 55 33)(29 34 56 44)(30 45 49 35)(31 36 50 46)(32 47 51 37)
(1 20 64 10)(2 17 57 15)(3 22 58 12)(4 19 59 9)(5 24 60 14)(6 21 61 11)(7 18 62 16)(8 23 63 13)(25 40 56 46)(26 37 49 43)(27 34 50 48)(28 39 51 45)(29 36 52 42)(30 33 53 47)(31 38 54 44)(32 35 55 41)
(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 38 20 54 64 44 10 31)(2 26 17 37 57 49 15 43)(3 46 22 25 58 40 12 56)(4 51 19 45 59 28 9 39)(5 34 24 50 60 48 14 27)(6 30 21 33 61 53 11 47)(7 42 18 29 62 36 16 52)(8 55 23 41 63 32 13 35)
G:=sub<Sym(64)| (1,16,60,22)(2,23,61,9)(3,10,62,24)(4,17,63,11)(5,12,64,18)(6,19,57,13)(7,14,58,20)(8,21,59,15)(25,38,52,48)(26,41,53,39)(27,40,54,42)(28,43,55,33)(29,34,56,44)(30,45,49,35)(31,36,50,46)(32,47,51,37), (1,20,64,10)(2,17,57,15)(3,22,58,12)(4,19,59,9)(5,24,60,14)(6,21,61,11)(7,18,62,16)(8,23,63,13)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,20,54,64,44,10,31)(2,26,17,37,57,49,15,43)(3,46,22,25,58,40,12,56)(4,51,19,45,59,28,9,39)(5,34,24,50,60,48,14,27)(6,30,21,33,61,53,11,47)(7,42,18,29,62,36,16,52)(8,55,23,41,63,32,13,35)>;
G:=Group( (1,16,60,22)(2,23,61,9)(3,10,62,24)(4,17,63,11)(5,12,64,18)(6,19,57,13)(7,14,58,20)(8,21,59,15)(25,38,52,48)(26,41,53,39)(27,40,54,42)(28,43,55,33)(29,34,56,44)(30,45,49,35)(31,36,50,46)(32,47,51,37), (1,20,64,10)(2,17,57,15)(3,22,58,12)(4,19,59,9)(5,24,60,14)(6,21,61,11)(7,18,62,16)(8,23,63,13)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,20,54,64,44,10,31)(2,26,17,37,57,49,15,43)(3,46,22,25,58,40,12,56)(4,51,19,45,59,28,9,39)(5,34,24,50,60,48,14,27)(6,30,21,33,61,53,11,47)(7,42,18,29,62,36,16,52)(8,55,23,41,63,32,13,35) );
G=PermutationGroup([[(1,16,60,22),(2,23,61,9),(3,10,62,24),(4,17,63,11),(5,12,64,18),(6,19,57,13),(7,14,58,20),(8,21,59,15),(25,38,52,48),(26,41,53,39),(27,40,54,42),(28,43,55,33),(29,34,56,44),(30,45,49,35),(31,36,50,46),(32,47,51,37)], [(1,20,64,10),(2,17,57,15),(3,22,58,12),(4,19,59,9),(5,24,60,14),(6,21,61,11),(7,18,62,16),(8,23,63,13),(25,40,56,46),(26,37,49,43),(27,34,50,48),(28,39,51,45),(29,36,52,42),(30,33,53,47),(31,38,54,44),(32,35,55,41)], [(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)], [(1,38,20,54,64,44,10,31),(2,26,17,37,57,49,15,43),(3,46,22,25,58,40,12,56),(4,51,19,45,59,28,9,39),(5,34,24,50,60,48,14,27),(6,30,21,33,61,53,11,47),(7,42,18,29,62,36,16,52),(8,55,23,41,63,32,13,35)]])
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 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4≀C2 | C4.10D4 | C42⋊C22 |
| kernel | C42.408D4 | C42.2C22 | C4×M4(2) | C42.6C4 | C2×C42.C2 | C2×C4⋊C4 | C42.C2 | C42 | C22×C4 | C22 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.408D4 ►in GL6(𝔽17)
| 1 | 9 | 0 | 0 | 0 | 0 |
| 13 | 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 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 4 | 2 |
| 0 | 0 | 10 | 11 | 0 | 13 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 9 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 2 | 7 |
| 0 | 0 | 14 | 6 | 9 | 7 |
| 0 | 0 | 6 | 7 | 1 | 14 |
| 0 | 0 | 8 | 9 | 4 | 13 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 16 | 0 |
| 0 | 0 | 13 | 6 | 13 | 15 |
| 0 | 0 | 9 | 11 | 9 | 13 |
| 0 | 0 | 13 | 9 | 6 | 2 |
G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,9,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,0,16,2,10,0,0,1,0,11,11,0,0,0,0,4,0,0,0,0,0,2,13],[5,9,0,0,0,0,10,12,0,0,0,0,0,0,14,14,6,8,0,0,3,6,7,9,0,0,2,9,1,4,0,0,7,7,14,13],[0,6,0,0,0,0,12,3,0,0,0,0,0,0,0,13,9,13,0,0,2,6,11,9,0,0,16,13,9,6,0,0,0,15,13,2] >;
C42.408D4 in GAP, Magma, Sage, TeX
C_4^2._{408}D_4 % in TeX
G:=Group("C4^2.408D4"); // GroupNames label
G:=SmallGroup(128,260);
// by ID
G=gap.SmallGroup(128,260);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations