p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.78D4, C42.165C23, (C4×D4).7C4, C4.57(C2×D8), (C2×C4).19D8, C4⋊D4.14C4, C4.D8⋊21C2, C4.77(C2×SD16), (C2×C4).28SD16, C4.10D8⋊33C2, C4⋊C8.259C22, C4.82(C8⋊C22), C42.106(C2×C4), (C22×C4).743D4, C4⋊Q8.238C22, C4.36(D4⋊C4), C4⋊M4(2)⋊20C2, C4.84(C8.C22), C4⋊1D4.127C22, C22.6(D4⋊C4), (C2×C42).209C22, C23.109(C22⋊C4), C2.12(C23.36D4), C22.26C24.14C2, C2.16(M4(2).8C22), (C2×C4⋊C8)⋊7C2, C4⋊C4.37(C2×C4), (C2×D4).31(C2×C4), C2.16(C2×D4⋊C4), (C2×C4).1236(C2×D4), (C2×C4).159(C22×C4), (C22×C4).231(C2×C4), (C2×C4).182(C22⋊C4), C22.223(C2×C22⋊C4), SmallGroup(128,279)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.78D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 292 in 130 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, 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, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, C4.D8, C4.10D8, C2×C4⋊C8, C4⋊M4(2), C22.26C24, C42.78D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C8⋊C22, C8.C22, M4(2).8C22, C2×D4⋊C4, C23.36D4, C42.78D4
►On 64 points
(1 45 5 41)(2 42 6 46)(3 47 7 43)(4 44 8 48)(9 57 13 61)(10 62 14 58)(11 59 15 63)(12 64 16 60)(17 33 21 37)(18 38 22 34)(19 35 23 39)(20 40 24 36)(25 56 29 52)(26 53 30 49)(27 50 31 54)(28 55 32 51)
(1 18 50 57)(2 58 51 19)(3 20 52 59)(4 60 53 21)(5 22 54 61)(6 62 55 23)(7 24 56 63)(8 64 49 17)(9 41 34 27)(10 28 35 42)(11 43 36 29)(12 30 37 44)(13 45 38 31)(14 32 39 46)(15 47 40 25)(16 26 33 48)
(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 64 18 49 50 17 57 8)(2 7 58 24 51 56 19 63)(3 62 20 55 52 23 59 6)(4 5 60 22 53 54 21 61)(9 48 41 16 34 26 27 33)(10 40 28 25 35 15 42 47)(11 46 43 14 36 32 29 39)(12 38 30 31 37 13 44 45)
G:=sub<Sym(64)| (1,45,5,41)(2,42,6,46)(3,47,7,43)(4,44,8,48)(9,57,13,61)(10,62,14,58)(11,59,15,63)(12,64,16,60)(17,33,21,37)(18,38,22,34)(19,35,23,39)(20,40,24,36)(25,56,29,52)(26,53,30,49)(27,50,31,54)(28,55,32,51), (1,18,50,57)(2,58,51,19)(3,20,52,59)(4,60,53,21)(5,22,54,61)(6,62,55,23)(7,24,56,63)(8,64,49,17)(9,41,34,27)(10,28,35,42)(11,43,36,29)(12,30,37,44)(13,45,38,31)(14,32,39,46)(15,47,40,25)(16,26,33,48), (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), (1,64,18,49,50,17,57,8)(2,7,58,24,51,56,19,63)(3,62,20,55,52,23,59,6)(4,5,60,22,53,54,21,61)(9,48,41,16,34,26,27,33)(10,40,28,25,35,15,42,47)(11,46,43,14,36,32,29,39)(12,38,30,31,37,13,44,45)>;
G:=Group( (1,45,5,41)(2,42,6,46)(3,47,7,43)(4,44,8,48)(9,57,13,61)(10,62,14,58)(11,59,15,63)(12,64,16,60)(17,33,21,37)(18,38,22,34)(19,35,23,39)(20,40,24,36)(25,56,29,52)(26,53,30,49)(27,50,31,54)(28,55,32,51), (1,18,50,57)(2,58,51,19)(3,20,52,59)(4,60,53,21)(5,22,54,61)(6,62,55,23)(7,24,56,63)(8,64,49,17)(9,41,34,27)(10,28,35,42)(11,43,36,29)(12,30,37,44)(13,45,38,31)(14,32,39,46)(15,47,40,25)(16,26,33,48), (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), (1,64,18,49,50,17,57,8)(2,7,58,24,51,56,19,63)(3,62,20,55,52,23,59,6)(4,5,60,22,53,54,21,61)(9,48,41,16,34,26,27,33)(10,40,28,25,35,15,42,47)(11,46,43,14,36,32,29,39)(12,38,30,31,37,13,44,45) );
G=PermutationGroup([[(1,45,5,41),(2,42,6,46),(3,47,7,43),(4,44,8,48),(9,57,13,61),(10,62,14,58),(11,59,15,63),(12,64,16,60),(17,33,21,37),(18,38,22,34),(19,35,23,39),(20,40,24,36),(25,56,29,52),(26,53,30,49),(27,50,31,54),(28,55,32,51)], [(1,18,50,57),(2,58,51,19),(3,20,52,59),(4,60,53,21),(5,22,54,61),(6,62,55,23),(7,24,56,63),(8,64,49,17),(9,41,34,27),(10,28,35,42),(11,43,36,29),(12,30,37,44),(13,45,38,31),(14,32,39,46),(15,47,40,25),(16,26,33,48)], [(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)], [(1,64,18,49,50,17,57,8),(2,7,58,24,51,56,19,63),(3,62,20,55,52,23,59,6),(4,5,60,22,53,54,21,61),(9,48,41,16,34,26,27,33),(10,40,28,25,35,15,42,47),(11,46,43,14,36,32,29,39),(12,38,30,31,37,13,44,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D8 | SD16 | C8⋊C22 | C8.C22 | M4(2).8C22 |
| kernel | C42.78D4 | C4.D8 | C4.10D8 | C2×C4⋊C8 | C4⋊M4(2) | C22.26C24 | C4×D4 | C4⋊D4 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of C42.78D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 8 | 0 |
| 0 | 0 | 0 | 4 | 8 | 9 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 2 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 10 | 14 | 6 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 11 | 3 | 13 | 13 |
| 0 | 0 | 11 | 3 | 10 | 10 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 10 | 14 | 6 |
| 0 | 0 | 5 | 3 | 3 | 9 |
| 0 | 0 | 0 | 10 | 7 | 10 |
| 0 | 0 | 0 | 7 | 4 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,4,4,0,0,0,0,8,8,0,4,0,0,0,9,4,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,2,16,1,0,0,16,16,0,16,0,0,0,0,0,16,0,0,0,0,1,0],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,11,0,11,11,0,0,10,0,3,3,0,0,14,8,13,10,0,0,6,3,13,10],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,11,5,0,0,0,0,10,3,10,7,0,0,14,3,7,4,0,0,6,9,10,13] >;
C42.78D4 in GAP, Magma, Sage, TeX
C_4^2._{78}D_4 % in TeX
G:=Group("C4^2.78D4"); // GroupNames label
G:=SmallGroup(128,279);
// by ID
G=gap.SmallGroup(128,279);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,184,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations