p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.91D4, C42.18Q8, C42.628C23, C4⋊C8.12C4, C8⋊1C8⋊27C2, C4.84(C2×D8), (C2×C4).132D8, C4.56(C2×Q16), (C2×C4).59Q16, C4.5(C2.D8), C4.44(C8○D4), C22⋊C8.11C4, (C4×C8).37C22, (C22×C4).30Q8, C4⋊C8.268C22, C23.53(C4⋊C4), C42.124(C2×C4), (C22×C4).748D4, C22.5(C2.D8), C4⋊M4(2).24C2, (C2×C42).227C22, C2.6(M4(2).C4), C42.12C4.33C2, C2.6(C42.6C22), (C2×C4⋊C8).20C2, C2.5(C2×C2.D8), (C2×C8).27(C2×C4), (C2×C4).35(C4⋊C4), C22.85(C2×C4⋊C4), (C2×C4).155(C2×Q8), (C2×C4).1464(C2×D4), (C22×C4).249(C2×C4), (C2×C4).510(C22×C4), SmallGroup(128,303)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.91D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c3 >
Subgroups: 132 in 84 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C8⋊1C8, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.91D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C8○D4, C2×D8, C2×Q16, C42.6C22, C2×C2.D8, M4(2).C4, C42.91D4
►On 64 points
(1 7 5 3)(2 56 6 52)(4 50 8 54)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(25 60 29 64)(26 32 30 28)(27 62 31 58)(33 39 37 35)(34 48 38 44)(36 42 40 46)(41 47 45 43)(49 55 53 51)(57 63 61 59)
(1 35 49 43)(2 36 50 44)(3 37 51 45)(4 38 52 46)(5 39 53 47)(6 40 54 48)(7 33 55 41)(8 34 56 42)(9 60 19 31)(10 61 20 32)(11 62 21 25)(12 63 22 26)(13 64 23 27)(14 57 24 28)(15 58 17 29)(16 59 18 30)
(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 29 35 15 49 58 43 17)(2 28 36 14 50 57 44 24)(3 27 37 13 51 64 45 23)(4 26 38 12 52 63 46 22)(5 25 39 11 53 62 47 21)(6 32 40 10 54 61 48 20)(7 31 33 9 55 60 41 19)(8 30 34 16 56 59 42 18)
G:=sub<Sym(64)| (1,7,5,3)(2,56,6,52)(4,50,8,54)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(25,60,29,64)(26,32,30,28)(27,62,31,58)(33,39,37,35)(34,48,38,44)(36,42,40,46)(41,47,45,43)(49,55,53,51)(57,63,61,59), (1,35,49,43)(2,36,50,44)(3,37,51,45)(4,38,52,46)(5,39,53,47)(6,40,54,48)(7,33,55,41)(8,34,56,42)(9,60,19,31)(10,61,20,32)(11,62,21,25)(12,63,22,26)(13,64,23,27)(14,57,24,28)(15,58,17,29)(16,59,18,30), (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), (1,29,35,15,49,58,43,17)(2,28,36,14,50,57,44,24)(3,27,37,13,51,64,45,23)(4,26,38,12,52,63,46,22)(5,25,39,11,53,62,47,21)(6,32,40,10,54,61,48,20)(7,31,33,9,55,60,41,19)(8,30,34,16,56,59,42,18)>;
G:=Group( (1,7,5,3)(2,56,6,52)(4,50,8,54)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(25,60,29,64)(26,32,30,28)(27,62,31,58)(33,39,37,35)(34,48,38,44)(36,42,40,46)(41,47,45,43)(49,55,53,51)(57,63,61,59), (1,35,49,43)(2,36,50,44)(3,37,51,45)(4,38,52,46)(5,39,53,47)(6,40,54,48)(7,33,55,41)(8,34,56,42)(9,60,19,31)(10,61,20,32)(11,62,21,25)(12,63,22,26)(13,64,23,27)(14,57,24,28)(15,58,17,29)(16,59,18,30), (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), (1,29,35,15,49,58,43,17)(2,28,36,14,50,57,44,24)(3,27,37,13,51,64,45,23)(4,26,38,12,52,63,46,22)(5,25,39,11,53,62,47,21)(6,32,40,10,54,61,48,20)(7,31,33,9,55,60,41,19)(8,30,34,16,56,59,42,18) );
G=PermutationGroup([[(1,7,5,3),(2,56,6,52),(4,50,8,54),(9,17,13,21),(10,16,14,12),(11,19,15,23),(18,24,22,20),(25,60,29,64),(26,32,30,28),(27,62,31,58),(33,39,37,35),(34,48,38,44),(36,42,40,46),(41,47,45,43),(49,55,53,51),(57,63,61,59)], [(1,35,49,43),(2,36,50,44),(3,37,51,45),(4,38,52,46),(5,39,53,47),(6,40,54,48),(7,33,55,41),(8,34,56,42),(9,60,19,31),(10,61,20,32),(11,62,21,25),(12,63,22,26),(13,64,23,27),(14,57,24,28),(15,58,17,29),(16,59,18,30)], [(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)], [(1,29,35,15,49,58,43,17),(2,28,36,14,50,57,44,24),(3,27,37,13,51,64,45,23),(4,26,38,12,52,63,46,22),(5,25,39,11,53,62,47,21),(6,32,40,10,54,61,48,20),(7,31,33,9,55,60,41,19),(8,30,34,16,56,59,42,18)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | D8 | Q16 | C8○D4 | M4(2).C4 |
| kernel | C42.91D4 | C8⋊1C8 | C2×C4⋊C8 | C4⋊M4(2) | C42.12C4 | C22⋊C8 | C4⋊C8 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 |
Matrix representation of C42.91D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 7 | 1 | 0 | 0 |
| 1 | 10 | 0 | 0 |
| 0 | 0 | 15 | 13 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,1,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[14,14,0,0,3,14,0,0,0,0,16,1,0,0,15,1],[7,1,0,0,1,10,0,0,0,0,15,0,0,0,13,2] >;
C42.91D4 in GAP, Magma, Sage, TeX
C_4^2._{91}D_4 % in TeX
G:=Group("C4^2.91D4"); // GroupNames label
G:=SmallGroup(128,303);
// by ID
G=gap.SmallGroup(128,303);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,723,1123,136,172]);
// 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=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations