p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.43Q8, C42.306D4, C8.16M4(2), C42.625C23, (C4×C8).20C4, C8⋊2C8⋊25C2, C8⋊1C8⋊26C2, (C22×C8).41C4, C4.123(C4○D8), (C22×C4).75Q8, C4⋊C8.216C22, C23.51(C4⋊C4), C42.312(C2×C4), (C4×C8).391C22, (C22×C4).542D4, C4.45(C2×M4(2)), C2.8(C4⋊M4(2)), C42.6C4.28C2, C22.5(C8.C4), (C2×C42).1043C22, C2.5(C23.25D4), (C2×C4×C8).42C2, (C2×C4).76(C4⋊C4), (C2×C8).232(C2×C4), C2.8(C2×C8.C4), C22.82(C2×C4⋊C4), (C2×C4).152(C2×Q8), (C2×C4).1461(C2×D4), (C2×C4).507(C22×C4), (C22×C4).476(C2×C4), SmallGroup(128,300)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.43Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c3 >
Subgroups: 124 in 84 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C8⋊2C8, C8⋊1C8, C2×C4×C8, C42.6C4, C42.43Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C8.C4, C2×C4⋊C4, C2×M4(2), C4○D8, C4⋊M4(2), C23.25D4, C2×C8.C4, C42.43Q8
►On 64 points
(1 7 5 3)(2 8 6 4)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)(17 61 21 57)(18 62 22 58)(19 63 23 59)(20 64 24 60)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)
(1 33 56 47)(2 34 49 48)(3 35 50 41)(4 36 51 42)(5 37 52 43)(6 38 53 44)(7 39 54 45)(8 40 55 46)(9 23 31 61)(10 24 32 62)(11 17 25 63)(12 18 26 64)(13 19 27 57)(14 20 28 58)(15 21 29 59)(16 22 30 60)
(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 32 39 60 52 14 41 18)(2 27 40 63 53 9 42 21)(3 30 33 58 54 12 43 24)(4 25 34 61 55 15 44 19)(5 28 35 64 56 10 45 22)(6 31 36 59 49 13 46 17)(7 26 37 62 50 16 47 20)(8 29 38 57 51 11 48 23)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52), (1,33,56,47)(2,34,49,48)(3,35,50,41)(4,36,51,42)(5,37,52,43)(6,38,53,44)(7,39,54,45)(8,40,55,46)(9,23,31,61)(10,24,32,62)(11,17,25,63)(12,18,26,64)(13,19,27,57)(14,20,28,58)(15,21,29,59)(16,22,30,60), (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), (1,32,39,60,52,14,41,18)(2,27,40,63,53,9,42,21)(3,30,33,58,54,12,43,24)(4,25,34,61,55,15,44,19)(5,28,35,64,56,10,45,22)(6,31,36,59,49,13,46,17)(7,26,37,62,50,16,47,20)(8,29,38,57,51,11,48,23)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52), (1,33,56,47)(2,34,49,48)(3,35,50,41)(4,36,51,42)(5,37,52,43)(6,38,53,44)(7,39,54,45)(8,40,55,46)(9,23,31,61)(10,24,32,62)(11,17,25,63)(12,18,26,64)(13,19,27,57)(14,20,28,58)(15,21,29,59)(16,22,30,60), (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), (1,32,39,60,52,14,41,18)(2,27,40,63,53,9,42,21)(3,30,33,58,54,12,43,24)(4,25,34,61,55,15,44,19)(5,28,35,64,56,10,45,22)(6,31,36,59,49,13,46,17)(7,26,37,62,50,16,47,20)(8,29,38,57,51,11,48,23) );
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28),(17,61,21,57),(18,62,22,58),(19,63,23,59),(20,64,24,60),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52)], [(1,33,56,47),(2,34,49,48),(3,35,50,41),(4,36,51,42),(5,37,52,43),(6,38,53,44),(7,39,54,45),(8,40,55,46),(9,23,31,61),(10,24,32,62),(11,17,25,63),(12,18,26,64),(13,19,27,57),(14,20,28,58),(15,21,29,59),(16,22,30,60)], [(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)], [(1,32,39,60,52,14,41,18),(2,27,40,63,53,9,42,21),(3,30,33,58,54,12,43,24),(4,25,34,61,55,15,44,19),(5,28,35,64,56,10,45,22),(6,31,36,59,49,13,46,17),(7,26,37,62,50,16,47,20),(8,29,38,57,51,11,48,23)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | M4(2) | C4○D8 | C8.C4 |
| kernel | C42.43Q8 | C8⋊2C8 | C8⋊1C8 | C2×C4×C8 | C42.6C4 | C4×C8 | C22×C8 | C42 | C42 | C22×C4 | C22×C4 | C8 | C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 8 | 8 |
Matrix representation of C42.43Q8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 8 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[8,0,0,0,0,2,0,0,0,0,4,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,0,13,0,0,1,0] >;
C42.43Q8 in GAP, Magma, Sage, TeX
C_4^2._{43}Q_8 % in TeX
G:=Group("C4^2.43Q8"); // GroupNames label
G:=SmallGroup(128,300);
// by ID
G=gap.SmallGroup(128,300);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,184,1123,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations