p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.262D4, C42.727C23, C8⋊3Q8⋊8C2, C4⋊Q16⋊21C2, C8.14(C4○D4), C8.D4.5C2, Q16⋊C4⋊18C2, C4⋊C4.124C23, (C4×C8).188C22, (C2×C8).464C23, (C2×C4).383C24, C4.SD16⋊45C2, C23.270(C2×D4), (C22×C4).481D4, (C4×M4(2)).6C2, C4⋊Q8.298C22, (C4×Q8).99C22, C4.Q8.34C22, C4.24(C8.C22), (C2×Q16).68C22, (C2×Q8).124C23, C8⋊C4.140C22, (C2×C42).869C22, C22.643(C22×D4), C22⋊Q8.184C22, (C22×C4).1061C23, Q8⋊C4.133C22, (C2×M4(2)).291C22, C23.37C23.36C2, C2.80(C22.26C24), C4.68(C2×C4○D4), (C2×C4).1226(C2×D4), C2.48(C2×C8.C22), SmallGroup(128,1917)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.262D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=a2c3 >
Subgroups: 292 in 179 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4.Q8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C2×Q16, C4×M4(2), Q16⋊C4, C8.D4, C4.SD16, C4⋊Q16, C8⋊3Q8, C23.37C23, C42.262D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, C22.26C24, C2×C8.C22, C42.262D4
►On 64 points
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 51 21 55)(18 56 22 52)(19 53 23 49)(20 50 24 54)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 63 45 59)(42 60 46 64)(43 57 47 61)(44 62 48 58)
(1 39 31 12)(2 36 32 9)(3 33 25 14)(4 38 26 11)(5 35 27 16)(6 40 28 13)(7 37 29 10)(8 34 30 15)(17 64 53 44)(18 61 54 41)(19 58 55 46)(20 63 56 43)(21 60 49 48)(22 57 50 45)(23 62 51 42)(24 59 52 47)
(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 48 5 44)(2 47 6 43)(3 46 7 42)(4 45 8 41)(9 52 13 56)(10 51 14 55)(11 50 15 54)(12 49 16 53)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(25 58 29 62)(26 57 30 61)(27 64 31 60)(28 63 32 59)
G:=sub<Sym(64)| (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,51,21,55)(18,56,22,52)(19,53,23,49)(20,50,24,54)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,63,45,59)(42,60,46,64)(43,57,47,61)(44,62,48,58), (1,39,31,12)(2,36,32,9)(3,33,25,14)(4,38,26,11)(5,35,27,16)(6,40,28,13)(7,37,29,10)(8,34,30,15)(17,64,53,44)(18,61,54,41)(19,58,55,46)(20,63,56,43)(21,60,49,48)(22,57,50,45)(23,62,51,42)(24,59,52,47), (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), (1,48,5,44)(2,47,6,43)(3,46,7,42)(4,45,8,41)(9,52,13,56)(10,51,14,55)(11,50,15,54)(12,49,16,53)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,58,29,62)(26,57,30,61)(27,64,31,60)(28,63,32,59)>;
G:=Group( (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,51,21,55)(18,56,22,52)(19,53,23,49)(20,50,24,54)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,63,45,59)(42,60,46,64)(43,57,47,61)(44,62,48,58), (1,39,31,12)(2,36,32,9)(3,33,25,14)(4,38,26,11)(5,35,27,16)(6,40,28,13)(7,37,29,10)(8,34,30,15)(17,64,53,44)(18,61,54,41)(19,58,55,46)(20,63,56,43)(21,60,49,48)(22,57,50,45)(23,62,51,42)(24,59,52,47), (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), (1,48,5,44)(2,47,6,43)(3,46,7,42)(4,45,8,41)(9,52,13,56)(10,51,14,55)(11,50,15,54)(12,49,16,53)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,58,29,62)(26,57,30,61)(27,64,31,60)(28,63,32,59) );
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,51,21,55),(18,56,22,52),(19,53,23,49),(20,50,24,54),(25,31,29,27),(26,28,30,32),(33,39,37,35),(34,36,38,40),(41,63,45,59),(42,60,46,64),(43,57,47,61),(44,62,48,58)], [(1,39,31,12),(2,36,32,9),(3,33,25,14),(4,38,26,11),(5,35,27,16),(6,40,28,13),(7,37,29,10),(8,34,30,15),(17,64,53,44),(18,61,54,41),(19,58,55,46),(20,63,56,43),(21,60,49,48),(22,57,50,45),(23,62,51,42),(24,59,52,47)], [(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)], [(1,48,5,44),(2,47,6,43),(3,46,7,42),(4,45,8,41),(9,52,13,56),(10,51,14,55),(11,50,15,54),(12,49,16,53),(17,39,21,35),(18,38,22,34),(19,37,23,33),(20,36,24,40),(25,58,29,62),(26,57,30,61),(27,64,31,60),(28,63,32,59)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4J | 4K | 4L | ··· | 4S | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8.C22 |
| kernel | C42.262D4 | C4×M4(2) | Q16⋊C4 | C8.D4 | C4.SD16 | C4⋊Q16 | C8⋊3Q8 | C23.37C23 | C42 | C22×C4 | C8 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 |
Matrix representation of C42.262D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 16 | 2 |
| 0 | 0 | 0 | 4 | 16 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 4 | 13 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 13 | 1 | 15 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 4 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 7 |
| 0 | 0 | 0 | 3 | 12 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,13,0,0,0,1,0,4,4,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,13,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,13,16,16,0,0,1,1,0,0,0,0,0,15,0,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,5,5,3,0,0,0,5,12,14,3,0,0,0,0,0,12,0,0,0,0,7,0] >;
C42.262D4 in GAP, Magma, Sage, TeX
C_4^2._{262}D_4 % in TeX
G:=Group("C4^2.262D4"); // GroupNames label
G:=SmallGroup(128,1917);
// by ID
G=gap.SmallGroup(128,1917);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,520,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations