p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8:7M4(2), C42.51Q8, C42.320D4, C42.624C23, (C4xC8).19C4, C8:1C8:25C2, (C2xC4).72D8, C4.83(C2xD8), (C2xC4).34Q16, C4.55(C2xQ16), (C22xC8).34C4, C4.14(C2.D8), C4.6(C8.C4), (C22xC4).82Q8, C4:C8.215C22, C23.50(C4:C4), C42.311(C2xC4), (C4xC8).390C22, (C22xC4).575D4, C4.44(C2xM4(2)), C22.12(C2.D8), C4:M4(2).21C2, C2.7(C4:M4(2)), (C2xC42).1042C22, (C2xC4xC8).32C2, C2.4(C2xC2.D8), (C2xC4).75(C4:C4), (C2xC8).220(C2xC4), C2.7(C2xC8.C4), C22.81(C2xC4:C4), (C2xC4).151(C2xQ8), (C2xC4).1460(C2xD4), (C2xC4).506(C22xC4), (C22xC4).475(C2xC4), SmallGroup(128,299)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:7M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >
Subgroups: 140 in 92 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), C22xC4, C4xC8, C4xC8, C4:C8, C4:C8, C2xC42, C22xC8, C2xM4(2), C8:1C8, C2xC4xC8, C4:M4(2), C8:7M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, M4(2), D8, Q16, C22xC4, C2xD4, C2xQ8, C2.D8, C8.C4, C2xC4:C4, C2xM4(2), C2xD8, C2xQ16, C4:M4(2), C2xC2.D8, C2xC8.C4, C8:7M4(2)
►On 64 points
(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 16 48 26 39 60 54 19)(2 15 41 25 40 59 55 18)(3 14 42 32 33 58 56 17)(4 13 43 31 34 57 49 24)(5 12 44 30 35 64 50 23)(6 11 45 29 36 63 51 22)(7 10 46 28 37 62 52 21)(8 9 47 27 38 61 53 20)
(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
G:=sub<Sym(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), (1,16,48,26,39,60,54,19)(2,15,41,25,40,59,55,18)(3,14,42,32,33,58,56,17)(4,13,43,31,34,57,49,24)(5,12,44,30,35,64,50,23)(6,11,45,29,36,63,51,22)(7,10,46,28,37,62,52,21)(8,9,47,27,38,61,53,20), (9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)>;
G:=Group( (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), (1,16,48,26,39,60,54,19)(2,15,41,25,40,59,55,18)(3,14,42,32,33,58,56,17)(4,13,43,31,34,57,49,24)(5,12,44,30,35,64,50,23)(6,11,45,29,36,63,51,22)(7,10,46,28,37,62,52,21)(8,9,47,27,38,61,53,20), (9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31) );
G=PermutationGroup([[(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)], [(1,16,48,26,39,60,54,19),(2,15,41,25,40,59,55,18),(3,14,42,32,33,58,56,17),(4,13,43,31,34,57,49,24),(5,12,44,30,35,64,50,23),(6,11,45,29,36,63,51,22),(7,10,46,28,37,62,52,21),(8,9,47,27,38,61,53,20)], [(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 |
| kernel | C8:7M4(2) | C8:1C8 | C2xC4xC8 | C4:M4(2) | C4xC8 | C22xC8 | C42 | C42 | C22xC4 | C22xC4 | C8 | C2xC4 | C2xC4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 8 |
Matrix representation of C8:7M4(2) ►in GL4(F17) generated by
| 8 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 15 | 2 |
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 14 | 15 |
| 0 | 0 | 4 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [8,0,0,0,0,15,0,0,0,0,9,15,0,0,0,2],[0,13,0,0,1,0,0,0,0,0,14,4,0,0,15,3],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;
C8:7M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes_7M_4(2)
% in TeX
G:=Group("C8:7M4(2)"); // GroupNames label
G:=SmallGroup(128,299);
// by ID
G=gap.SmallGroup(128,299);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,1430,1123,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations