p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.2C42, M5(2)⋊9C4, C23.30SD16, (C2×C8).6Q8, C8.13(C4⋊C4), (C2×C4).9Q16, C8.C4⋊3C4, (C2×C8).338D4, (C2×C4).123D8, C2.D8.17C4, C4.3(C2.D8), (C2×C4).22SD16, C8.37(C22⋊C4), (C22×C4).193D4, C4.33(D4⋊C4), C22.3(C4.Q8), C4.15(Q8⋊C4), C2.2(M5(2)⋊C2), C2.2(C8.17D4), (C2×M5(2)).15C2, C4.8(C2.C42), (C22×C8).206C22, C22.3(Q8⋊C4), C22.47(D4⋊C4), C2.17(C22.4Q16), (C2×C8).53(C2×C4), (C2×C4).29(C4⋊C4), (C2×C2.D8).31C2, (C2×C8.C4).5C2, (C2×C4).231(C22⋊C4), SmallGroup(128,119)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.2C42
G = < a,b,c | a8=b4=1, c4=a2, bab-1=a-1, cac-1=a5, cbc-1=a3b >
Subgroups: 136 in 68 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2.D8, C2.D8, C8.C4, C8.C4, C2×C16, M5(2), M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C2×C2.D8, C2×C8.C4, C2×M5(2), C8.2C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, M5(2)⋊C2, C8.17D4, C8.2C42
►On 64 points
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 60 54 64 58 52 62 56)
(1 24 63 40)(2 19 64 35)(3 22 49 38)(4 17 50 33)(5 20 51 36)(6 31 52 47)(7 18 53 34)(8 29 54 45)(9 32 55 48)(10 27 56 43)(11 30 57 46)(12 25 58 41)(13 28 59 44)(14 23 60 39)(15 26 61 42)(16 21 62 37)
(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)
G:=sub<Sym(64)| (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,60,54,64,58,52,62,56), (1,24,63,40)(2,19,64,35)(3,22,49,38)(4,17,50,33)(5,20,51,36)(6,31,52,47)(7,18,53,34)(8,29,54,45)(9,32,55,48)(10,27,56,43)(11,30,57,46)(12,25,58,41)(13,28,59,44)(14,23,60,39)(15,26,61,42)(16,21,62,37), (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)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,60,54,64,58,52,62,56), (1,24,63,40)(2,19,64,35)(3,22,49,38)(4,17,50,33)(5,20,51,36)(6,31,52,47)(7,18,53,34)(8,29,54,45)(9,32,55,48)(10,27,56,43)(11,30,57,46)(12,25,58,41)(13,28,59,44)(14,23,60,39)(15,26,61,42)(16,21,62,37), (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) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,20,22,24,26,28,30,32),(33,43,37,47,41,35,45,39),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,60,54,64,58,52,62,56)], [(1,24,63,40),(2,19,64,35),(3,22,49,38),(4,17,50,33),(5,20,51,36),(6,31,52,47),(7,18,53,34),(8,29,54,45),(9,32,55,48),(10,27,56,43),(11,30,57,46),(12,25,58,41),(13,28,59,44),(14,23,60,39),(15,26,61,42),(16,21,62,37)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | SD16 | M5(2)⋊C2 | C8.17D4 |
| kernel | C8.2C42 | C2×C2.D8 | C2×C8.C4 | C2×M5(2) | C2.D8 | C8.C4 | M5(2) | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of C8.2C42 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 15 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[6,11,0,0,0,0,9,11,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[6,15,0,0,0,0,9,11,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,1,0,0,0,0,0,0,1,0,0] >;
C8.2C42 in GAP, Magma, Sage, TeX
C_8._2C_4^2
% in TeX
G:=Group("C8.2C4^2"); // GroupNames label
G:=SmallGroup(128,119);
// by ID
G=gap.SmallGroup(128,119);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;
// generators/relations