p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.283D4, C42.733C23, C4.552- 1+4, Q8.Q8⋊1C2, C8⋊3Q8⋊23C2, C8⋊2Q8⋊15C2, C4.31(C4○D8), C4⋊C4.170C23, C4⋊C8.314C22, (C2×C4).429C24, (C2×C8).333C23, (C4×C8).269C22, C4.SD16⋊28C2, (C22×C4).512D4, C23.297(C2×D4), C4⋊Q8.313C22, C2.D8.38C22, C4.Q8.86C22, C4.30(C8.C22), (C2×Q8).163C23, (C4×Q8).110C22, Q8⋊C4.6C22, C22⋊C8.196C22, (C2×C42).890C22, C23.20D4.1C2, C22.689(C22×D4), C22⋊Q8.203C22, C42.12C4.38C2, (C22×C4).1094C23, C42.C2.130C22, C42⋊C2.164C22, C23.37C23.40C2, C2.77(C23.38C23), C2.46(C2×C4○D8), (C2×C4).554(C2×D4), C2.61(C2×C8.C22), SmallGroup(128,1963)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.283D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a2c3 >
Subgroups: 268 in 163 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C42.12C4, Q8.Q8, C23.20D4, C4.SD16, C8⋊3Q8, C8⋊2Q8, C23.37C23, C42.283D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C8.C22, C22×D4, 2- 1+4, C23.38C23, C2×C4○D8, C2×C8.C22, C42.283D4
►On 64 points
(1 47 15 27)(2 48 16 28)(3 41 9 29)(4 42 10 30)(5 43 11 31)(6 44 12 32)(7 45 13 25)(8 46 14 26)(17 53 61 37)(18 54 62 38)(19 55 63 39)(20 56 64 40)(21 49 57 33)(22 50 58 34)(23 51 59 35)(24 52 60 36)
(1 13 5 9)(2 8 6 4)(3 15 7 11)(10 16 14 12)(17 63 21 59)(18 20 22 24)(19 57 23 61)(25 43 29 47)(26 32 30 28)(27 45 31 41)(33 51 37 55)(34 36 38 40)(35 53 39 49)(42 48 46 44)(50 52 54 56)(58 60 62 64)
(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 54 5 50)(2 33 6 37)(3 52 7 56)(4 39 8 35)(9 36 13 40)(10 55 14 51)(11 34 15 38)(12 53 16 49)(17 28 21 32)(18 43 22 47)(19 26 23 30)(20 41 24 45)(25 64 29 60)(27 62 31 58)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (1,47,15,27)(2,48,16,28)(3,41,9,29)(4,42,10,30)(5,43,11,31)(6,44,12,32)(7,45,13,25)(8,46,14,26)(17,53,61,37)(18,54,62,38)(19,55,63,39)(20,56,64,40)(21,49,57,33)(22,50,58,34)(23,51,59,35)(24,52,60,36), (1,13,5,9)(2,8,6,4)(3,15,7,11)(10,16,14,12)(17,63,21,59)(18,20,22,24)(19,57,23,61)(25,43,29,47)(26,32,30,28)(27,45,31,41)(33,51,37,55)(34,36,38,40)(35,53,39,49)(42,48,46,44)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,33,6,37)(3,52,7,56)(4,39,8,35)(9,36,13,40)(10,55,14,51)(11,34,15,38)(12,53,16,49)(17,28,21,32)(18,43,22,47)(19,26,23,30)(20,41,24,45)(25,64,29,60)(27,62,31,58)(42,63,46,59)(44,61,48,57)>;
G:=Group( (1,47,15,27)(2,48,16,28)(3,41,9,29)(4,42,10,30)(5,43,11,31)(6,44,12,32)(7,45,13,25)(8,46,14,26)(17,53,61,37)(18,54,62,38)(19,55,63,39)(20,56,64,40)(21,49,57,33)(22,50,58,34)(23,51,59,35)(24,52,60,36), (1,13,5,9)(2,8,6,4)(3,15,7,11)(10,16,14,12)(17,63,21,59)(18,20,22,24)(19,57,23,61)(25,43,29,47)(26,32,30,28)(27,45,31,41)(33,51,37,55)(34,36,38,40)(35,53,39,49)(42,48,46,44)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,33,6,37)(3,52,7,56)(4,39,8,35)(9,36,13,40)(10,55,14,51)(11,34,15,38)(12,53,16,49)(17,28,21,32)(18,43,22,47)(19,26,23,30)(20,41,24,45)(25,64,29,60)(27,62,31,58)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([[(1,47,15,27),(2,48,16,28),(3,41,9,29),(4,42,10,30),(5,43,11,31),(6,44,12,32),(7,45,13,25),(8,46,14,26),(17,53,61,37),(18,54,62,38),(19,55,63,39),(20,56,64,40),(21,49,57,33),(22,50,58,34),(23,51,59,35),(24,52,60,36)], [(1,13,5,9),(2,8,6,4),(3,15,7,11),(10,16,14,12),(17,63,21,59),(18,20,22,24),(19,57,23,61),(25,43,29,47),(26,32,30,28),(27,45,31,41),(33,51,37,55),(34,36,38,40),(35,53,39,49),(42,48,46,44),(50,52,54,56),(58,60,62,64)], [(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)], [(1,54,5,50),(2,33,6,37),(3,52,7,56),(4,39,8,35),(9,36,13,40),(10,55,14,51),(11,34,15,38),(12,53,16,49),(17,28,21,32),(18,43,22,47),(19,26,23,30),(20,41,24,45),(25,64,29,60),(27,62,31,58),(42,63,46,59),(44,61,48,57)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | C8.C22 | 2- 1+4 |
| kernel | C42.283D4 | C42.12C4 | Q8.Q8 | C23.20D4 | C4.SD16 | C8⋊3Q8 | C8⋊2Q8 | C23.37C23 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.283D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 2 | 10 | 16 | 7 |
| 0 | 0 | 11 | 7 | 7 | 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 | 13 | 0 | 16 | 0 |
| 0 | 0 | 4 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 8 |
| 0 | 0 | 13 | 0 | 16 | 1 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 16 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,13,2,11,0,0,12,9,10,7,0,0,0,0,16,7,0,0,0,0,7,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,13,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,0,0,0,0,0,0,15,0,0,0,0,0,0,1,13,0,4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,8,1,1,16],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,8,1,1,16,0,0,0,1,0,0] >;
C42.283D4 in GAP, Magma, Sage, TeX
C_4^2._{283}D_4 % in TeX
G:=Group("C4^2.283D4"); // GroupNames label
G:=SmallGroup(128,1963);
// by ID
G=gap.SmallGroup(128,1963);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations