p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊2M5(2), C42.12C8, C4⋊C16⋊13C2, (C4×C8).38C4, C4.13(C4⋊C8), C8.37(C4⋊C4), C8.41(C2×Q8), (C2×C8).61Q8, (C2×C8).288D4, C8.134(C2×D4), C22.6(C4⋊C8), (C22×C8).31C4, (C2×C42).48C4, C23.37(C2×C8), (C22×C4).15C8, C2.7(C2×M5(2)), (C2×C16).52C22, C42.328(C2×C4), (C2×C8).627C23, (C4×C8).373C22, (C2×C4).82M4(2), C4.65(C2×M4(2)), (C2×M5(2)).23C2, C22.48(C22×C8), (C22×C8).578C22, (C2×C4×C8).66C2, C2.13(C2×C4⋊C8), C4.79(C2×C4⋊C4), (C2×C4).88(C2×C8), (C2×C8).251(C2×C4), (C2×C4).137(C4⋊C4), (C2×C4).612(C22×C4), (C22×C4).487(C2×C4), SmallGroup(128,882)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊M5(2)
G = < a,b,c | a4=b16=c2=1, bab-1=a-1, ac=ca, cbc=b9 >
Subgroups: 108 in 86 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, M5(2), C2×C42, C22×C8, C4⋊C16, C2×C4×C8, C2×M5(2), C4⋊M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, C2×M5(2), C4⋊M5(2)
►On 64 points
(1 32 64 42)(2 43 49 17)(3 18 50 44)(4 45 51 19)(5 20 52 46)(6 47 53 21)(7 22 54 48)(8 33 55 23)(9 24 56 34)(10 35 57 25)(11 26 58 36)(12 37 59 27)(13 28 60 38)(14 39 61 29)(15 30 62 40)(16 41 63 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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)
G:=sub<Sym(64)| (1,32,64,42)(2,43,49,17)(3,18,50,44)(4,45,51,19)(5,20,52,46)(6,47,53,21)(7,22,54,48)(8,33,55,23)(9,24,56,34)(10,35,57,25)(11,26,58,36)(12,37,59,27)(13,28,60,38)(14,39,61,29)(15,30,62,40)(16,41,63,31), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)>;
G:=Group( (1,32,64,42)(2,43,49,17)(3,18,50,44)(4,45,51,19)(5,20,52,46)(6,47,53,21)(7,22,54,48)(8,33,55,23)(9,24,56,34)(10,35,57,25)(11,26,58,36)(12,37,59,27)(13,28,60,38)(14,39,61,29)(15,30,62,40)(16,41,63,31), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63) );
G=PermutationGroup([[(1,32,64,42),(2,43,49,17),(3,18,50,44),(4,45,51,19),(5,20,52,46),(6,47,53,21),(7,22,54,48),(8,33,55,23),(9,24,56,34),(10,35,57,25),(11,26,58,36),(12,37,59,27),(13,28,60,38),(14,39,61,29),(15,30,62,40),(16,41,63,31)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(33,41),(35,43),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | M5(2) |
| kernel | C4⋊M5(2) | C4⋊C16 | C2×C4×C8 | C2×M5(2) | C4×C8 | C2×C42 | C22×C8 | C42 | C22×C4 | C2×C8 | C2×C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 16 |
Matrix representation of C4⋊M5(2) ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,1,0,0,0,0,0,0,9,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C4⋊M5(2) in GAP, Magma, Sage, TeX
C_4\rtimes M_5(2)
% in TeX
G:=Group("C4:M5(2)"); // GroupNames label
G:=SmallGroup(128,882);
// by ID
G=gap.SmallGroup(128,882);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1430,102,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^9>;
// generators/relations