p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.52D4, C42.616C23, D4⋊C8⋊32C2, Q8⋊C8⋊36C2, C4.4(C8○D4), C4⋊D4.7C4, (C4×C8).8C22, C22⋊Q8.7C4, C42.66(C2×C4), C4.4D4.6C4, (C4×D4).7C22, C42.C2.8C4, (C4×Q8).7C22, C4⋊C8.200C22, (C22×C4).206D4, C4.135(C8⋊C22), C4⋊M4(2)⋊18C2, C42.12C4⋊13C2, C4.129(C8.C22), C23.48(C22⋊C4), (C2×C42).169C22, C2.7(C23.36D4), C2.12(C42⋊C22), C23.36C23.4C2, C4⋊C4.56(C2×C4), (C2×D4).54(C2×C4), (C2×Q8).49(C2×C4), (C2×C4).1452(C2×D4), (C2×C4).82(C22⋊C4), (C2×C4).321(C22×C4), (C22×C4).191(C2×C4), C22.171(C2×C22⋊C4), C2.21((C22×C8)⋊C2), SmallGroup(128,227)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.52D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 212 in 108 conjugacy classes, 46 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, 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, C22⋊C8, 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, C2×M4(2), D4⋊C8, Q8⋊C8, C4⋊M4(2), C42.12C4, C23.36C23, C42.52D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8○D4, C8⋊C22, C8.C22, (C22×C8)⋊C2, C23.36D4, C42⋊C22, C42.52D4
►On 64 points
(1 18 51 9)(2 10 52 19)(3 20 53 11)(4 12 54 21)(5 22 55 13)(6 14 56 23)(7 24 49 15)(8 16 50 17)(25 42 62 34)(26 35 63 43)(27 44 64 36)(28 37 57 45)(29 46 58 38)(30 39 59 47)(31 48 60 40)(32 33 61 41)
(1 28 55 61)(2 29 56 62)(3 30 49 63)(4 31 50 64)(5 32 51 57)(6 25 52 58)(7 26 53 59)(8 27 54 60)(9 45 22 33)(10 46 23 34)(11 47 24 35)(12 48 17 36)(13 41 18 37)(14 42 19 38)(15 43 20 39)(16 44 21 40)
(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 54 55 60 61 8)(2 53 29 59 56 7 62 26)(3 58 30 6 49 25 63 52)(4 5 31 32 50 51 64 57)(9 40 45 16 22 44 33 21)(10 15 46 43 23 20 34 39)(11 42 47 19 24 38 35 14)(12 18 48 37 17 13 36 41)
G:=sub<Sym(64)| (1,18,51,9)(2,10,52,19)(3,20,53,11)(4,12,54,21)(5,22,55,13)(6,14,56,23)(7,24,49,15)(8,16,50,17)(25,42,62,34)(26,35,63,43)(27,44,64,36)(28,37,57,45)(29,46,58,38)(30,39,59,47)(31,48,60,40)(32,33,61,41), (1,28,55,61)(2,29,56,62)(3,30,49,63)(4,31,50,64)(5,32,51,57)(6,25,52,58)(7,26,53,59)(8,27,54,60)(9,45,22,33)(10,46,23,34)(11,47,24,35)(12,48,17,36)(13,41,18,37)(14,42,19,38)(15,43,20,39)(16,44,21,40), (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), (1,27,28,54,55,60,61,8)(2,53,29,59,56,7,62,26)(3,58,30,6,49,25,63,52)(4,5,31,32,50,51,64,57)(9,40,45,16,22,44,33,21)(10,15,46,43,23,20,34,39)(11,42,47,19,24,38,35,14)(12,18,48,37,17,13,36,41)>;
G:=Group( (1,18,51,9)(2,10,52,19)(3,20,53,11)(4,12,54,21)(5,22,55,13)(6,14,56,23)(7,24,49,15)(8,16,50,17)(25,42,62,34)(26,35,63,43)(27,44,64,36)(28,37,57,45)(29,46,58,38)(30,39,59,47)(31,48,60,40)(32,33,61,41), (1,28,55,61)(2,29,56,62)(3,30,49,63)(4,31,50,64)(5,32,51,57)(6,25,52,58)(7,26,53,59)(8,27,54,60)(9,45,22,33)(10,46,23,34)(11,47,24,35)(12,48,17,36)(13,41,18,37)(14,42,19,38)(15,43,20,39)(16,44,21,40), (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), (1,27,28,54,55,60,61,8)(2,53,29,59,56,7,62,26)(3,58,30,6,49,25,63,52)(4,5,31,32,50,51,64,57)(9,40,45,16,22,44,33,21)(10,15,46,43,23,20,34,39)(11,42,47,19,24,38,35,14)(12,18,48,37,17,13,36,41) );
G=PermutationGroup([[(1,18,51,9),(2,10,52,19),(3,20,53,11),(4,12,54,21),(5,22,55,13),(6,14,56,23),(7,24,49,15),(8,16,50,17),(25,42,62,34),(26,35,63,43),(27,44,64,36),(28,37,57,45),(29,46,58,38),(30,39,59,47),(31,48,60,40),(32,33,61,41)], [(1,28,55,61),(2,29,56,62),(3,30,49,63),(4,31,50,64),(5,32,51,57),(6,25,52,58),(7,26,53,59),(8,27,54,60),(9,45,22,33),(10,46,23,34),(11,47,24,35),(12,48,17,36),(13,41,18,37),(14,42,19,38),(15,43,20,39),(16,44,21,40)], [(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)], [(1,27,28,54,55,60,61,8),(2,53,29,59,56,7,62,26),(3,58,30,6,49,25,63,52),(4,5,31,32,50,51,64,57),(9,40,45,16,22,44,33,21),(10,15,46,43,23,20,34,39),(11,42,47,19,24,38,35,14),(12,18,48,37,17,13,36,41)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C8○D4 | C8⋊C22 | C8.C22 | C42⋊C22 |
| kernel | C42.52D4 | D4⋊C8 | Q8⋊C8 | C4⋊M4(2) | C42.12C4 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C42.C2 | C42 | C22×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.52D4 ►in GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 16 | 15 |
| 0 | 0 | 0 | 4 | 15 | 15 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 1 | 16 | 13 | 13 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 15 | 4 | 4 |
| 0 | 0 | 5 | 0 | 4 | 0 |
| 0 | 0 | 13 | 12 | 12 | 12 |
| 0 | 0 | 12 | 0 | 2 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 15 | 4 | 4 |
| 0 | 0 | 5 | 13 | 4 | 8 |
| 0 | 0 | 13 | 12 | 12 | 12 |
| 0 | 0 | 12 | 14 | 2 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,4,4,1,16,0,0,16,15,0,13,0,0,15,15,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,5,13,12,0,0,15,0,12,0,0,0,4,4,12,2,0,0,4,0,12,0],[8,9,0,0,0,0,0,9,0,0,0,0,0,0,5,5,13,12,0,0,15,13,12,14,0,0,4,4,12,2,0,0,4,8,12,4] >;
C42.52D4 in GAP, Magma, Sage, TeX
C_4^2._{52}D_4 % in TeX
G:=Group("C4^2.52D4"); // GroupNames label
G:=SmallGroup(128,227);
// by ID
G=gap.SmallGroup(128,227);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1059,1123,570,136,172]);
// 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,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations