p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.400D4, C42.605C23, C4.22C4≀C2, C4⋊Q8.8C4, D4⋊C8⋊25C2, C4⋊D4.4C4, C4⋊1D4.7C4, C4.28(C8○D4), C42.58(C2×C4), (C4×D4).4C22, C4⋊C8.251C22, (C4×M4(2))⋊13C2, (C4×C8).310C22, (C22×C4).202D4, C4.132(C8⋊C22), C42.12C4⋊10C2, C23.44(C22⋊C4), (C2×C42).161C22, C2.4(C23.37D4), C22.26C24.2C2, C2.7(C2×C4≀C2), C4⋊C4.51(C2×C4), (C2×D4).51(C2×C4), (C2×C4).1447(C2×D4), (C2×C4).310(C22×C4), (C22×C4).183(C2×C4), (C2×C4).165(C22⋊C4), C22.160(C2×C22⋊C4), C2.16((C22×C8)⋊C2), SmallGroup(128,216)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.400D4
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=a2b-1c3 >
Subgroups: 284 in 131 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, D4⋊C8, C4×M4(2), C42.12C4, C22.26C24, C42.400D4
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, C23.37D4, C2×C4≀C2, C42.400D4
►On 64 points
(1 41 22 40)(2 46 23 37)(3 43 24 34)(4 48 17 39)(5 45 18 36)(6 42 19 33)(7 47 20 38)(8 44 21 35)(9 27 58 51)(10 32 59 56)(11 29 60 53)(12 26 61 50)(13 31 62 55)(14 28 63 52)(15 25 64 49)(16 30 57 54)
(1 58 18 13)(2 59 19 14)(3 60 20 15)(4 61 21 16)(5 62 22 9)(6 63 23 10)(7 64 24 11)(8 57 17 12)(25 43 53 38)(26 44 54 39)(27 45 55 40)(28 46 56 33)(29 47 49 34)(30 48 50 35)(31 41 51 36)(32 42 52 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 39 58 26 18 44 13 54)(2 25 59 43 19 53 14 38)(3 42 60 52 20 37 15 32)(4 51 61 36 21 31 16 41)(5 35 62 30 22 48 9 50)(6 29 63 47 23 49 10 34)(7 46 64 56 24 33 11 28)(8 55 57 40 17 27 12 45)
G:=sub<Sym(64)| (1,41,22,40)(2,46,23,37)(3,43,24,34)(4,48,17,39)(5,45,18,36)(6,42,19,33)(7,47,20,38)(8,44,21,35)(9,27,58,51)(10,32,59,56)(11,29,60,53)(12,26,61,50)(13,31,62,55)(14,28,63,52)(15,25,64,49)(16,30,57,54), (1,58,18,13)(2,59,19,14)(3,60,20,15)(4,61,21,16)(5,62,22,9)(6,63,23,10)(7,64,24,11)(8,57,17,12)(25,43,53,38)(26,44,54,39)(27,45,55,40)(28,46,56,33)(29,47,49,34)(30,48,50,35)(31,41,51,36)(32,42,52,37), (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), (1,39,58,26,18,44,13,54)(2,25,59,43,19,53,14,38)(3,42,60,52,20,37,15,32)(4,51,61,36,21,31,16,41)(5,35,62,30,22,48,9,50)(6,29,63,47,23,49,10,34)(7,46,64,56,24,33,11,28)(8,55,57,40,17,27,12,45)>;
G:=Group( (1,41,22,40)(2,46,23,37)(3,43,24,34)(4,48,17,39)(5,45,18,36)(6,42,19,33)(7,47,20,38)(8,44,21,35)(9,27,58,51)(10,32,59,56)(11,29,60,53)(12,26,61,50)(13,31,62,55)(14,28,63,52)(15,25,64,49)(16,30,57,54), (1,58,18,13)(2,59,19,14)(3,60,20,15)(4,61,21,16)(5,62,22,9)(6,63,23,10)(7,64,24,11)(8,57,17,12)(25,43,53,38)(26,44,54,39)(27,45,55,40)(28,46,56,33)(29,47,49,34)(30,48,50,35)(31,41,51,36)(32,42,52,37), (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), (1,39,58,26,18,44,13,54)(2,25,59,43,19,53,14,38)(3,42,60,52,20,37,15,32)(4,51,61,36,21,31,16,41)(5,35,62,30,22,48,9,50)(6,29,63,47,23,49,10,34)(7,46,64,56,24,33,11,28)(8,55,57,40,17,27,12,45) );
G=PermutationGroup([[(1,41,22,40),(2,46,23,37),(3,43,24,34),(4,48,17,39),(5,45,18,36),(6,42,19,33),(7,47,20,38),(8,44,21,35),(9,27,58,51),(10,32,59,56),(11,29,60,53),(12,26,61,50),(13,31,62,55),(14,28,63,52),(15,25,64,49),(16,30,57,54)], [(1,58,18,13),(2,59,19,14),(3,60,20,15),(4,61,21,16),(5,62,22,9),(6,63,23,10),(7,64,24,11),(8,57,17,12),(25,43,53,38),(26,44,54,39),(27,45,55,40),(28,46,56,33),(29,47,49,34),(30,48,50,35),(31,41,51,36),(32,42,52,37)], [(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)], [(1,39,58,26,18,44,13,54),(2,25,59,43,19,53,14,38),(3,42,60,52,20,37,15,32),(4,51,61,36,21,31,16,41),(5,35,62,30,22,48,9,50),(6,29,63,47,23,49,10,34),(7,46,64,56,24,33,11,28),(8,55,57,40,17,27,12,45)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C8○D4 | C8⋊C22 |
| kernel | C42.400D4 | D4⋊C8 | C4×M4(2) | C42.12C4 | C22.26C24 | C4⋊D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.400D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 9 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 10 | 14 |
| 0 | 15 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[9,0,0,0,0,8,0,0,0,0,0,10,0,0,14,14],[0,15,0,0,15,0,0,0,0,0,0,10,0,0,3,0] >;
C42.400D4 in GAP, Magma, Sage, TeX
C_4^2._{400}D_4 % in TeX
G:=Group("C4^2.400D4"); // GroupNames label
G:=SmallGroup(128,216);
// by ID
G=gap.SmallGroup(128,216);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1059,184,1123,570,136,172]);
// 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=a^2*b^-1*c^3>;
// generators/relations