p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.47D4, C42.604C23, D4⋊C8⋊24C2, Q8⋊C8⋊28C2, C22.6C4≀C2, C4⋊D4.3C4, C22⋊Q8.3C4, C4.27(C8○D4), C42.57(C2×C4), C4.4D4.4C4, (C4×D4).3C22, C42.C2.6C4, (C4×Q8).3C22, C4⋊C8.250C22, (C4×M4(2))⋊12C2, (C4×C8).309C22, (C22×C4).657D4, C4.131(C8⋊C22), C4.125(C8.C22), C23.96(C22⋊C4), (C2×C42).160C22, C2.5(C23.36D4), C23.36C23.2C2, (C2×C4⋊C8)⋊4C2, C2.6(C2×C4≀C2), C4⋊C4.50(C2×C4), (C2×D4).50(C2×C4), (C2×Q8).45(C2×C4), (C2×C4).1137(C2×D4), (C2×C4).77(C22⋊C4), (C2×C4).309(C22×C4), (C22×C4).182(C2×C4), C22.159(C2×C22⋊C4), C2.15((C22×C8)⋊C2), SmallGroup(128,215)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.47D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 220 in 117 conjugacy classes, 48 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), D4⋊C8, Q8⋊C8, C4×M4(2), C2×C4⋊C8, C23.36C23, C42.47D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8○D4, C8⋊C22, C8.C22, (C22×C8)⋊C2, C23.36D4, C2×C4≀C2, C42.47D4
►On 64 points
(1 18 52 10)(2 23 53 15)(3 20 54 12)(4 17 55 9)(5 22 56 14)(6 19 49 11)(7 24 50 16)(8 21 51 13)(25 35 62 46)(26 40 63 43)(27 37 64 48)(28 34 57 45)(29 39 58 42)(30 36 59 47)(31 33 60 44)(32 38 61 41)
(1 28 56 61)(2 29 49 62)(3 30 50 63)(4 31 51 64)(5 32 52 57)(6 25 53 58)(7 26 54 59)(8 27 55 60)(9 44 21 37)(10 45 22 38)(11 46 23 39)(12 47 24 40)(13 48 17 33)(14 41 18 34)(15 42 19 35)(16 43 20 36)
(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 27 28 55 56 60 61 8)(2 54 29 59 49 7 62 26)(3 58 30 6 50 25 63 53)(4 5 31 32 51 52 64 57)(9 22 44 38 21 10 37 45)(11 16 46 43 23 20 39 36)(12 42 47 19 24 35 40 15)(13 18 48 34 17 14 33 41)
G:=sub<Sym(64)| (1,18,52,10)(2,23,53,15)(3,20,54,12)(4,17,55,9)(5,22,56,14)(6,19,49,11)(7,24,50,16)(8,21,51,13)(25,35,62,46)(26,40,63,43)(27,37,64,48)(28,34,57,45)(29,39,58,42)(30,36,59,47)(31,33,60,44)(32,38,61,41), (1,28,56,61)(2,29,49,62)(3,30,50,63)(4,31,51,64)(5,32,52,57)(6,25,53,58)(7,26,54,59)(8,27,55,60)(9,44,21,37)(10,45,22,38)(11,46,23,39)(12,47,24,40)(13,48,17,33)(14,41,18,34)(15,42,19,35)(16,43,20,36), (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), (1,27,28,55,56,60,61,8)(2,54,29,59,49,7,62,26)(3,58,30,6,50,25,63,53)(4,5,31,32,51,52,64,57)(9,22,44,38,21,10,37,45)(11,16,46,43,23,20,39,36)(12,42,47,19,24,35,40,15)(13,18,48,34,17,14,33,41)>;
G:=Group( (1,18,52,10)(2,23,53,15)(3,20,54,12)(4,17,55,9)(5,22,56,14)(6,19,49,11)(7,24,50,16)(8,21,51,13)(25,35,62,46)(26,40,63,43)(27,37,64,48)(28,34,57,45)(29,39,58,42)(30,36,59,47)(31,33,60,44)(32,38,61,41), (1,28,56,61)(2,29,49,62)(3,30,50,63)(4,31,51,64)(5,32,52,57)(6,25,53,58)(7,26,54,59)(8,27,55,60)(9,44,21,37)(10,45,22,38)(11,46,23,39)(12,47,24,40)(13,48,17,33)(14,41,18,34)(15,42,19,35)(16,43,20,36), (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), (1,27,28,55,56,60,61,8)(2,54,29,59,49,7,62,26)(3,58,30,6,50,25,63,53)(4,5,31,32,51,52,64,57)(9,22,44,38,21,10,37,45)(11,16,46,43,23,20,39,36)(12,42,47,19,24,35,40,15)(13,18,48,34,17,14,33,41) );
G=PermutationGroup([[(1,18,52,10),(2,23,53,15),(3,20,54,12),(4,17,55,9),(5,22,56,14),(6,19,49,11),(7,24,50,16),(8,21,51,13),(25,35,62,46),(26,40,63,43),(27,37,64,48),(28,34,57,45),(29,39,58,42),(30,36,59,47),(31,33,60,44),(32,38,61,41)], [(1,28,56,61),(2,29,49,62),(3,30,50,63),(4,31,51,64),(5,32,52,57),(6,25,53,58),(7,26,54,59),(8,27,55,60),(9,44,21,37),(10,45,22,38),(11,46,23,39),(12,47,24,40),(13,48,17,33),(14,41,18,34),(15,42,19,35),(16,43,20,36)], [(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)], [(1,27,28,55,56,60,61,8),(2,54,29,59,49,7,62,26),(3,58,30,6,50,25,63,53),(4,5,31,32,51,52,64,57),(9,22,44,38,21,10,37,45),(11,16,46,43,23,20,39,36),(12,42,47,19,24,35,40,15),(13,18,48,34,17,14,33,41)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C8○D4 | C4≀C2 | C8⋊C22 | C8.C22 |
| kernel | C42.47D4 | D4⋊C8 | Q8⋊C8 | C4×M4(2) | C2×C4⋊C8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C42.C2 | C42 | C22×C4 | C4 | C22 | C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 1 | 1 |
Matrix representation of C42.47D4 ►in GL4(𝔽17) generated by
| 13 | 2 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 2 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 8 | 13 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 12 | 0 |
| 8 | 13 | 0 | 0 |
| 15 | 9 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 12 | 5 |
G:=sub<GL(4,GF(17))| [13,1,0,0,2,4,0,0,0,0,1,2,0,0,16,16],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[8,0,0,0,13,9,0,0,0,0,12,12,0,0,11,0],[8,15,0,0,13,9,0,0,0,0,12,12,0,0,11,5] >;
C42.47D4 in GAP, Magma, Sage, TeX
C_4^2._{47}D_4 % in TeX
G:=Group("C4^2.47D4"); // GroupNames label
G:=SmallGroup(128,215);
// by ID
G=gap.SmallGroup(128,215);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1059,1123,570,136,172]);
// 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,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations