p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.265D4, C42.728C23, C4.992+ 1+4, C8⋊4D4⋊9C2, D4⋊D4⋊3C2, C8⋊5D4⋊21C2, D4.2D4⋊1C2, C4.28(C4○D8), C4.4D8⋊26C2, (C4×C8).73C22, C4⋊C8.311C22, C4⋊C4.146C23, C4.26(C8⋊C22), (C2×C4).405C24, (C2×C8).159C23, C4.SD16⋊11C2, (C2×D8).23C22, C23.284(C2×D4), (C22×C4).495D4, C4⋊Q8.300C22, (C2×D4).155C23, (C4×D4).104C22, (C2×Q8).142C23, Q8⋊C4.2C22, C42.12C4⋊33C2, C4⋊D4.188C22, C4⋊1D4.161C22, C22⋊C8.192C22, (C2×C42).872C22, (C2×SD16).84C22, C22.665(C22×D4), D4⋊C4.106C22, (C22×C4).1076C23, C22.26C24⋊17C2, C4.4D4.149C22, C2.76(C22.29C24), C2.41(C2×C4○D8), (C2×C4).537(C2×D4), C2.54(C2×C8⋊C22), (C2×C4○D4).171C22, SmallGroup(128,1939)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.265D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=b2c3 >
Subgroups: 492 in 219 conjugacy classes, 88 normal (34 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, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×C4○D4, C2×C4○D4, C42.12C4, D4⋊D4, D4.2D4, C4.4D8, C4.SD16, C8⋊5D4, C8⋊4D4, C22.26C24, C42.265D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C8⋊C22, C22×D4, 2+ 1+4, C22.29C24, C2×C4○D8, C2×C8⋊C22, C42.265D4
►On 64 points
(1 61 29 54)(2 62 30 55)(3 63 31 56)(4 64 32 49)(5 57 25 50)(6 58 26 51)(7 59 27 52)(8 60 28 53)(9 19 40 41)(10 20 33 42)(11 21 34 43)(12 22 35 44)(13 23 36 45)(14 24 37 46)(15 17 38 47)(16 18 39 48)
(1 39 5 35)(2 9 6 13)(3 33 7 37)(4 11 8 15)(10 27 14 31)(12 29 16 25)(17 64 21 60)(18 50 22 54)(19 58 23 62)(20 52 24 56)(26 36 30 40)(28 38 32 34)(41 51 45 55)(42 59 46 63)(43 53 47 49)(44 61 48 57)
(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)
(2 8)(3 7)(4 6)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 19)(20 24)(21 23)(26 32)(27 31)(28 30)(41 47)(42 46)(43 45)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)
G:=sub<Sym(64)| (1,61,29,54)(2,62,30,55)(3,63,31,56)(4,64,32,49)(5,57,25,50)(6,58,26,51)(7,59,27,52)(8,60,28,53)(9,19,40,41)(10,20,33,42)(11,21,34,43)(12,22,35,44)(13,23,36,45)(14,24,37,46)(15,17,38,47)(16,18,39,48), (1,39,5,35)(2,9,6,13)(3,33,7,37)(4,11,8,15)(10,27,14,31)(12,29,16,25)(17,64,21,60)(18,50,22,54)(19,58,23,62)(20,52,24,56)(26,36,30,40)(28,38,32,34)(41,51,45,55)(42,59,46,63)(43,53,47,49)(44,61,48,57), (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), (2,8)(3,7)(4,6)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(41,47)(42,46)(43,45)(49,58)(50,57)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)>;
G:=Group( (1,61,29,54)(2,62,30,55)(3,63,31,56)(4,64,32,49)(5,57,25,50)(6,58,26,51)(7,59,27,52)(8,60,28,53)(9,19,40,41)(10,20,33,42)(11,21,34,43)(12,22,35,44)(13,23,36,45)(14,24,37,46)(15,17,38,47)(16,18,39,48), (1,39,5,35)(2,9,6,13)(3,33,7,37)(4,11,8,15)(10,27,14,31)(12,29,16,25)(17,64,21,60)(18,50,22,54)(19,58,23,62)(20,52,24,56)(26,36,30,40)(28,38,32,34)(41,51,45,55)(42,59,46,63)(43,53,47,49)(44,61,48,57), (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), (2,8)(3,7)(4,6)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(41,47)(42,46)(43,45)(49,58)(50,57)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59) );
G=PermutationGroup([[(1,61,29,54),(2,62,30,55),(3,63,31,56),(4,64,32,49),(5,57,25,50),(6,58,26,51),(7,59,27,52),(8,60,28,53),(9,19,40,41),(10,20,33,42),(11,21,34,43),(12,22,35,44),(13,23,36,45),(14,24,37,46),(15,17,38,47),(16,18,39,48)], [(1,39,5,35),(2,9,6,13),(3,33,7,37),(4,11,8,15),(10,27,14,31),(12,29,16,25),(17,64,21,60),(18,50,22,54),(19,58,23,62),(20,52,24,56),(26,36,30,40),(28,38,32,34),(41,51,45,55),(42,59,46,63),(43,53,47,49),(44,61,48,57)], [(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)], [(2,8),(3,7),(4,6),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(17,19),(20,24),(21,23),(26,32),(27,31),(28,30),(41,47),(42,46),(43,45),(49,58),(50,57),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | 4L | 4M | 4N | 4O | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | C8⋊C22 | 2+ 1+4 |
| kernel | C42.265D4 | C42.12C4 | D4⋊D4 | D4.2D4 | C4.4D8 | C4.SD16 | C8⋊5D4 | C8⋊4D4 | C22.26C24 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.265D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 14 |
| 0 | 0 | 12 | 0 | 14 | 0 |
| 0 | 0 | 0 | 3 | 0 | 5 |
| 0 | 0 | 3 | 0 | 5 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,3,0,0,12,0,3,0,0,0,0,14,0,5,0,0,14,0,5,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C42.265D4 in GAP, Magma, Sage, TeX
C_4^2._{265}D_4 % in TeX
G:=Group("C4^2.265D4"); // GroupNames label
G:=SmallGroup(128,1939);
// by ID
G=gap.SmallGroup(128,1939);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=b^2*c^3>;
// generators/relations