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, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, M5(2), C2×C42, C22×C8, C4×C16, C16⋊5C4, C2×C4×C8, C2×M5(2), C4×M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, M5(2), C2×C42, C22×C8, C2×C4×C8, C2×M5(2), C4×M5(2)
►On 64 points
(1 63 20 36)(2 64 21 37)(3 49 22 38)(4 50 23 39)(5 51 24 40)(6 52 25 41)(7 53 26 42)(8 54 27 43)(9 55 28 44)(10 56 29 45)(11 57 30 46)(12 58 31 47)(13 59 32 48)(14 60 17 33)(15 61 18 34)(16 62 19 35)
(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 20)(2 29)(3 22)(4 31)(5 24)(6 17)(7 26)(8 19)(9 28)(10 21)(11 30)(12 23)(13 32)(14 25)(15 18)(16 27)(33 52)(34 61)(35 54)(36 63)(37 56)(38 49)(39 58)(40 51)(41 60)(42 53)(43 62)(44 55)(45 64)(46 57)(47 50)(48 59)
G:=sub<Sym(64)| (1,63,20,36)(2,64,21,37)(3,49,22,38)(4,50,23,39)(5,51,24,40)(6,52,25,41)(7,53,26,42)(8,54,27,43)(9,55,28,44)(10,56,29,45)(11,57,30,46)(12,58,31,47)(13,59,32,48)(14,60,17,33)(15,61,18,34)(16,62,19,35), (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), (1,20)(2,29)(3,22)(4,31)(5,24)(6,17)(7,26)(8,19)(9,28)(10,21)(11,30)(12,23)(13,32)(14,25)(15,18)(16,27)(33,52)(34,61)(35,54)(36,63)(37,56)(38,49)(39,58)(40,51)(41,60)(42,53)(43,62)(44,55)(45,64)(46,57)(47,50)(48,59)>;
G:=Group( (1,63,20,36)(2,64,21,37)(3,49,22,38)(4,50,23,39)(5,51,24,40)(6,52,25,41)(7,53,26,42)(8,54,27,43)(9,55,28,44)(10,56,29,45)(11,57,30,46)(12,58,31,47)(13,59,32,48)(14,60,17,33)(15,61,18,34)(16,62,19,35), (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), (1,20)(2,29)(3,22)(4,31)(5,24)(6,17)(7,26)(8,19)(9,28)(10,21)(11,30)(12,23)(13,32)(14,25)(15,18)(16,27)(33,52)(34,61)(35,54)(36,63)(37,56)(38,49)(39,58)(40,51)(41,60)(42,53)(43,62)(44,55)(45,64)(46,57)(47,50)(48,59) );
G=PermutationGroup([[(1,63,20,36),(2,64,21,37),(3,49,22,38),(4,50,23,39),(5,51,24,40),(6,52,25,41),(7,53,26,42),(8,54,27,43),(9,55,28,44),(10,56,29,45),(11,57,30,46),(12,58,31,47),(13,59,32,48),(14,60,17,33),(15,61,18,34),(16,62,19,35)], [(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)], [(1,20),(2,29),(3,22),(4,31),(5,24),(6,17),(7,26),(8,19),(9,28),(10,21),(11,30),(12,23),(13,32),(14,25),(15,18),(16,27),(33,52),(34,61),(35,54),(36,63),(37,56),(38,49),(39,58),(40,51),(41,60),(42,53),(43,62),(44,55),(45,64),(46,57),(47,50),(48,59)]])
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