p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.406D4, C42.144C23, C4.27C4≀C2, C4⋊Q8.15C4, C42.85(C2×C4), C42⋊C2.3C4, (C22×C4).224D4, (C4×M4(2)).18C2, C8⋊C4.144C22, C4.23(C4.10D4), C23.57(C22⋊C4), (C2×C42).188C22, C42.C2.93C22, C4○2(C42.2C22), C42.2C22⋊16C2, C23.37C23.9C2, C2.31(C2×C4≀C2), C4⋊C4.24(C2×C4), (C2×C4).1172(C2×D4), (C22×C4).210(C2×C4), (C2×C4).138(C22×C4), C2.10(C2×C4.10D4), (C2×C4).244(C22⋊C4), C22.202(C2×C22⋊C4), (C2×C4)○(C42.2C22), SmallGroup(128,258)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.406D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 196 in 113 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C42.2C22, C4×M4(2), C23.37C23, C42.406D4
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, C42.406D4
►On 64 points
(1 62 19 11)(2 63 20 12)(3 64 21 13)(4 57 22 14)(5 58 23 15)(6 59 24 16)(7 60 17 9)(8 61 18 10)(25 48 49 37)(26 41 50 38)(27 42 51 39)(28 43 52 40)(29 44 53 33)(30 45 54 34)(31 46 55 35)(32 47 56 36)
(1 60 23 13)(2 57 24 10)(3 62 17 15)(4 59 18 12)(5 64 19 9)(6 61 20 14)(7 58 21 11)(8 63 22 16)(25 39 53 46)(26 36 54 43)(27 33 55 48)(28 38 56 45)(29 35 49 42)(30 40 50 47)(31 37 51 44)(32 34 52 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 37 60 51 23 44 13 31)(2 26 57 36 24 54 10 43)(3 46 62 25 17 39 15 53)(4 56 59 45 18 28 12 38)(5 33 64 55 19 48 9 27)(6 30 61 40 20 50 14 47)(7 42 58 29 21 35 11 49)(8 52 63 41 22 32 16 34)
G:=sub<Sym(64)| (1,62,19,11)(2,63,20,12)(3,64,21,13)(4,57,22,14)(5,58,23,15)(6,59,24,16)(7,60,17,9)(8,61,18,10)(25,48,49,37)(26,41,50,38)(27,42,51,39)(28,43,52,40)(29,44,53,33)(30,45,54,34)(31,46,55,35)(32,47,56,36), (1,60,23,13)(2,57,24,10)(3,62,17,15)(4,59,18,12)(5,64,19,9)(6,61,20,14)(7,58,21,11)(8,63,22,16)(25,39,53,46)(26,36,54,43)(27,33,55,48)(28,38,56,45)(29,35,49,42)(30,40,50,47)(31,37,51,44)(32,34,52,41), (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), (1,37,60,51,23,44,13,31)(2,26,57,36,24,54,10,43)(3,46,62,25,17,39,15,53)(4,56,59,45,18,28,12,38)(5,33,64,55,19,48,9,27)(6,30,61,40,20,50,14,47)(7,42,58,29,21,35,11,49)(8,52,63,41,22,32,16,34)>;
G:=Group( (1,62,19,11)(2,63,20,12)(3,64,21,13)(4,57,22,14)(5,58,23,15)(6,59,24,16)(7,60,17,9)(8,61,18,10)(25,48,49,37)(26,41,50,38)(27,42,51,39)(28,43,52,40)(29,44,53,33)(30,45,54,34)(31,46,55,35)(32,47,56,36), (1,60,23,13)(2,57,24,10)(3,62,17,15)(4,59,18,12)(5,64,19,9)(6,61,20,14)(7,58,21,11)(8,63,22,16)(25,39,53,46)(26,36,54,43)(27,33,55,48)(28,38,56,45)(29,35,49,42)(30,40,50,47)(31,37,51,44)(32,34,52,41), (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), (1,37,60,51,23,44,13,31)(2,26,57,36,24,54,10,43)(3,46,62,25,17,39,15,53)(4,56,59,45,18,28,12,38)(5,33,64,55,19,48,9,27)(6,30,61,40,20,50,14,47)(7,42,58,29,21,35,11,49)(8,52,63,41,22,32,16,34) );
G=PermutationGroup([[(1,62,19,11),(2,63,20,12),(3,64,21,13),(4,57,22,14),(5,58,23,15),(6,59,24,16),(7,60,17,9),(8,61,18,10),(25,48,49,37),(26,41,50,38),(27,42,51,39),(28,43,52,40),(29,44,53,33),(30,45,54,34),(31,46,55,35),(32,47,56,36)], [(1,60,23,13),(2,57,24,10),(3,62,17,15),(4,59,18,12),(5,64,19,9),(6,61,20,14),(7,58,21,11),(8,63,22,16),(25,39,53,46),(26,36,54,43),(27,33,55,48),(28,38,56,45),(29,35,49,42),(30,40,50,47),(31,37,51,44),(32,34,52,41)], [(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)], [(1,37,60,51,23,44,13,31),(2,26,57,36,24,54,10,43),(3,46,62,25,17,39,15,53),(4,56,59,45,18,28,12,38),(5,33,64,55,19,48,9,27),(6,30,61,40,20,50,14,47),(7,42,58,29,21,35,11,49),(8,52,63,41,22,32,16,34)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4≀C2 | C4.10D4 |
| kernel | C42.406D4 | C42.2C22 | C4×M4(2) | C23.37C23 | C42⋊C2 | C4⋊Q8 | C42 | C22×C4 | C4 | C4 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 2 | 2 | 16 | 2 |
Matrix representation of C42.406D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[0,1,0,0,4,0,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,0,4,0,0,16,0] >;
C42.406D4 in GAP, Magma, Sage, TeX
C_4^2._{406}D_4 % in TeX
G:=Group("C4^2.406D4"); // GroupNames label
G:=SmallGroup(128,258);
// by ID
G=gap.SmallGroup(128,258);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,352,1123,1018,248,1971,102]);
// 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,a*c=c*a,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