p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.401D4, C42.606C23, C4.23C4≀C2, C4⋊Q8.9C4, Q8⋊C8⋊29C2, C22⋊Q8.4C4, C4.29(C8○D4), C42.59(C2×C4), (C4×Q8).4C22, C4⋊C8.252C22, (C4×C8).311C22, (C22×C4).203D4, (C4×M4(2)).16C2, C4.126(C8.C22), C23.45(C22⋊C4), (C2×C42).162C22, C42.12C4.17C2, C2.4(C23.38D4), C23.37C23.2C2, C2.8(C2×C4≀C2), C4⋊C4.52(C2×C4), (C2×Q8).46(C2×C4), (C2×C4).1448(C2×D4), (C22×C4).184(C2×C4), (C2×C4).311(C22×C4), (C2×C4).166(C22⋊C4), C22.161(C2×C22⋊C4), C2.17((C22×C8)⋊C2), SmallGroup(128,217)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.401D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, ad=da, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 188 in 109 conjugacy classes, 48 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, M4(2), C22×C4, C2×Q8, C2×Q8, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), Q8⋊C8, C4×M4(2), C42.12C4, C23.37C23, C42.401D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8○D4, C8.C22, (C22×C8)⋊C2, C23.38D4, C2×C4≀C2, C42.401D4
►On 64 points
(1 23 59 15)(2 20 60 12)(3 17 61 9)(4 22 62 14)(5 19 63 11)(6 24 64 16)(7 21 57 13)(8 18 58 10)(25 44 56 37)(26 41 49 34)(27 46 50 39)(28 43 51 36)(29 48 52 33)(30 45 53 38)(31 42 54 35)(32 47 55 40)
(1 17 63 13)(2 18 64 14)(3 19 57 15)(4 20 58 16)(5 21 59 9)(6 22 60 10)(7 23 61 11)(8 24 62 12)(25 39 52 42)(26 40 53 43)(27 33 54 44)(28 34 55 45)(29 35 56 46)(30 36 49 47)(31 37 50 48)(32 38 51 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 33 17 54 63 44 13 27)(2 30 18 36 64 49 14 47)(3 42 19 25 57 39 15 52)(4 55 20 45 58 28 16 34)(5 37 21 50 59 48 9 31)(6 26 22 40 60 53 10 43)(7 46 23 29 61 35 11 56)(8 51 24 41 62 32 12 38)
G:=sub<Sym(64)| (1,23,59,15)(2,20,60,12)(3,17,61,9)(4,22,62,14)(5,19,63,11)(6,24,64,16)(7,21,57,13)(8,18,58,10)(25,44,56,37)(26,41,49,34)(27,46,50,39)(28,43,51,36)(29,48,52,33)(30,45,53,38)(31,42,54,35)(32,47,55,40), (1,17,63,13)(2,18,64,14)(3,19,57,15)(4,20,58,16)(5,21,59,9)(6,22,60,10)(7,23,61,11)(8,24,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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), (1,33,17,54,63,44,13,27)(2,30,18,36,64,49,14,47)(3,42,19,25,57,39,15,52)(4,55,20,45,58,28,16,34)(5,37,21,50,59,48,9,31)(6,26,22,40,60,53,10,43)(7,46,23,29,61,35,11,56)(8,51,24,41,62,32,12,38)>;
G:=Group( (1,23,59,15)(2,20,60,12)(3,17,61,9)(4,22,62,14)(5,19,63,11)(6,24,64,16)(7,21,57,13)(8,18,58,10)(25,44,56,37)(26,41,49,34)(27,46,50,39)(28,43,51,36)(29,48,52,33)(30,45,53,38)(31,42,54,35)(32,47,55,40), (1,17,63,13)(2,18,64,14)(3,19,57,15)(4,20,58,16)(5,21,59,9)(6,22,60,10)(7,23,61,11)(8,24,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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), (1,33,17,54,63,44,13,27)(2,30,18,36,64,49,14,47)(3,42,19,25,57,39,15,52)(4,55,20,45,58,28,16,34)(5,37,21,50,59,48,9,31)(6,26,22,40,60,53,10,43)(7,46,23,29,61,35,11,56)(8,51,24,41,62,32,12,38) );
G=PermutationGroup([[(1,23,59,15),(2,20,60,12),(3,17,61,9),(4,22,62,14),(5,19,63,11),(6,24,64,16),(7,21,57,13),(8,18,58,10),(25,44,56,37),(26,41,49,34),(27,46,50,39),(28,43,51,36),(29,48,52,33),(30,45,53,38),(31,42,54,35),(32,47,55,40)], [(1,17,63,13),(2,18,64,14),(3,19,57,15),(4,20,58,16),(5,21,59,9),(6,22,60,10),(7,23,61,11),(8,24,62,12),(25,39,52,42),(26,40,53,43),(27,33,54,44),(28,34,55,45),(29,35,56,46),(30,36,49,47),(31,37,50,48),(32,38,51,41)], [(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)], [(1,33,17,54,63,44,13,27),(2,30,18,36,64,49,14,47),(3,42,19,25,57,39,15,52),(4,55,20,45,58,28,16,34),(5,37,21,50,59,48,9,31),(6,26,22,40,60,53,10,43),(7,46,23,29,61,35,11,56),(8,51,24,41,62,32,12,38)]])
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 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4≀C2 | C8○D4 | C8.C22 |
| kernel | C42.401D4 | Q8⋊C8 | C4×M4(2) | C42.12C4 | C23.37C23 | C22⋊Q8 | C4⋊Q8 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.401D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 4 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[0,9,0,0,9,0,0,0,0,0,16,0,0,0,0,4],[9,0,0,0,0,9,0,0,0,0,0,16,0,0,13,0] >;
C42.401D4 in GAP, Magma, Sage, TeX
C_4^2._{401}D_4 % in TeX
G:=Group("C4^2.401D4"); // GroupNames label
G:=SmallGroup(128,217);
// by ID
G=gap.SmallGroup(128,217);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1059,184,1123,570,136,172]);
// 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*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations