direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×M5(2), C8○2M5(2), C42.10C8, C8.17C42, C16⋊9(C2×C4), C4.5(C4×C8), (C4×C16)⋊14C2, (C4×C8).37C4, (C2×C8).18C8, C8.25(C2×C8), (C2×C8)○M5(2), C8○(C2×M5(2)), C4○2(C16⋊5C4), C8○2(C16⋊5C4), C16⋊5C4⋊14C2, C22.5(C4×C8), (C22×C8).46C4, (C2×C42).46C4, (C22×C4).10C8, C4.36(C2×C42), (C2×C4).73C42, C23.34(C2×C8), C8.66(C22×C4), C4.35(C22×C8), C2.2(C2×M5(2)), C42○(C2×M5(2)), C42○(C16⋊5C4), (C2×C8).621C23, C42.340(C2×C4), (C4×C8).442C22, (C2×C16).104C22, (C2×M5(2)).27C2, C22.24(C22×C8), (C22×C8).575C22, (C2×C4×C8).65C2, C2.11(C2×C4×C8), (C2×C4).99(C2×C8), (C4×C8)○(C2×M5(2)), (C4×C8)○(C16⋊5C4), (C2×C8).192(C2×C4), (C2×C4).606(C22×C4), (C22×C4).484(C2×C4), SmallGroup(128,839)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×M5(2)
G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Subgroups: 108 in 98 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C16 [×8], C42 [×2], C42 [×2], C2×C8 [×4], C2×C8 [×8], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C16 [×4], M5(2) [×8], C2×C42, C22×C8 [×2], C4×C16 [×2], C16⋊5C4 [×2], C2×C4×C8, C2×M5(2) [×2], C4×M5(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], M5(2) [×4], C2×C42, C22×C8 [×2], C2×C4×C8, C2×M5(2) [×2], C4×M5(2)
►On 64 points
(1 34 51 32)(2 35 52 17)(3 36 53 18)(4 37 54 19)(5 38 55 20)(6 39 56 21)(7 40 57 22)(8 41 58 23)(9 42 59 24)(10 43 60 25)(11 44 61 26)(12 45 62 27)(13 46 63 28)(14 47 64 29)(15 48 49 30)(16 33 50 31)
(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 51)(2 60)(3 53)(4 62)(5 55)(6 64)(7 57)(8 50)(9 59)(10 52)(11 61)(12 54)(13 63)(14 56)(15 49)(16 58)(17 43)(18 36)(19 45)(20 38)(21 47)(22 40)(23 33)(24 42)(25 35)(26 44)(27 37)(28 46)(29 39)(30 48)(31 41)(32 34)
G:=sub<Sym(64)| (1,34,51,32)(2,35,52,17)(3,36,53,18)(4,37,54,19)(5,38,55,20)(6,39,56,21)(7,40,57,22)(8,41,58,23)(9,42,59,24)(10,43,60,25)(11,44,61,26)(12,45,62,27)(13,46,63,28)(14,47,64,29)(15,48,49,30)(16,33,50,31), (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), (1,51)(2,60)(3,53)(4,62)(5,55)(6,64)(7,57)(8,50)(9,59)(10,52)(11,61)(12,54)(13,63)(14,56)(15,49)(16,58)(17,43)(18,36)(19,45)(20,38)(21,47)(22,40)(23,33)(24,42)(25,35)(26,44)(27,37)(28,46)(29,39)(30,48)(31,41)(32,34)>;
G:=Group( (1,34,51,32)(2,35,52,17)(3,36,53,18)(4,37,54,19)(5,38,55,20)(6,39,56,21)(7,40,57,22)(8,41,58,23)(9,42,59,24)(10,43,60,25)(11,44,61,26)(12,45,62,27)(13,46,63,28)(14,47,64,29)(15,48,49,30)(16,33,50,31), (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), (1,51)(2,60)(3,53)(4,62)(5,55)(6,64)(7,57)(8,50)(9,59)(10,52)(11,61)(12,54)(13,63)(14,56)(15,49)(16,58)(17,43)(18,36)(19,45)(20,38)(21,47)(22,40)(23,33)(24,42)(25,35)(26,44)(27,37)(28,46)(29,39)(30,48)(31,41)(32,34) );
G=PermutationGroup([(1,34,51,32),(2,35,52,17),(3,36,53,18),(4,37,54,19),(5,38,55,20),(6,39,56,21),(7,40,57,22),(8,41,58,23),(9,42,59,24),(10,43,60,25),(11,44,61,26),(12,45,62,27),(13,46,63,28),(14,47,64,29),(15,48,49,30),(16,33,50,31)], [(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)], [(1,51),(2,60),(3,53),(4,62),(5,55),(6,64),(7,57),(8,50),(9,59),(10,52),(11,61),(12,54),(13,63),(14,56),(15,49),(16,58),(17,43),(18,36),(19,45),(20,38),(21,47),(22,40),(23,33),(24,42),(25,35),(26,44),(27,37),(28,46),(29,39),(30,48),(31,41),(32,34)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) |
| kernel | C4×M5(2) | C4×C16 | C16⋊5C4 | C2×C4×C8 | C2×M5(2) | C4×C8 | M5(2) | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 16 | 2 | 2 | 8 | 16 | 8 | 16 |
Matrix representation of C4×M5(2) ►in GL3(𝔽17) generated by
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 13 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 2 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [4,0,0,0,1,0,0,0,1],[13,0,0,0,0,2,0,1,0],[1,0,0,0,1,0,0,0,16] >;
C4×M5(2) in GAP, Magma, Sage, TeX
C_4\times M_5(2)
% in TeX
G:=Group("C4xM5(2)"); // GroupNames label
G:=SmallGroup(128,839);
// by ID
G=gap.SmallGroup(128,839);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,1430,136,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations