p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊2M4(2), C42.649C23, C8⋊C8⋊7C2, Q8⋊C8⋊42C2, C8⋊1C8⋊18C2, C4.Q8.6C4, D4⋊C8.16C2, C8⋊4Q8⋊31C2, (C2×C8).183D4, C4.16(C8○D4), C8⋊6D4.14C2, C2.9(C8⋊6D4), (C4×SD16).4C2, (C2×SD16).2C4, D4⋊C4.11C4, C4⋊C8.227C22, (C4×C8).318C22, Q8⋊C4.11C4, (C4×D4).15C22, C22.140(C4×D4), C4.10(C2×M4(2)), (C4×Q8).15C22, C2.10(C8.26D4), C4.146(C8⋊C22), C2.7(SD16⋊C4), C4.140(C8.C22), C4⋊C4.62(C2×C4), (C2×C8).57(C2×C4), (C2×D4).60(C2×C4), (C2×Q8).55(C2×C4), (C2×C4).1485(C2×D4), (C2×C4).510(C4○D4), (C2×C4).341(C22×C4), SmallGroup(128,324)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b5 >
Subgroups: 152 in 81 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C8⋊1C8, C8⋊6D4, C4×SD16, C8⋊4Q8, C8⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8.C22, C8⋊6D4, SD16⋊C4, C8.26D4, C8⋊M4(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 31 61 15 38 22 56 45)(2 30 62 14 39 21 49 44)(3 29 63 13 40 20 50 43)(4 28 64 12 33 19 51 42)(5 27 57 11 34 18 52 41)(6 26 58 10 35 17 53 48)(7 25 59 9 36 24 54 47)(8 32 60 16 37 23 55 46)
(2 4)(3 7)(6 8)(9 43)(10 46)(11 41)(12 44)(13 47)(14 42)(15 45)(16 48)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(33 39)(35 37)(36 40)(49 51)(50 54)(53 55)(58 60)(59 63)(62 64)
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), (1,31,61,15,38,22,56,45)(2,30,62,14,39,21,49,44)(3,29,63,13,40,20,50,43)(4,28,64,12,33,19,51,42)(5,27,57,11,34,18,52,41)(6,26,58,10,35,17,53,48)(7,25,59,9,36,24,54,47)(8,32,60,16,37,23,55,46), (2,4)(3,7)(6,8)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(33,39)(35,37)(36,40)(49,51)(50,54)(53,55)(58,60)(59,63)(62,64)>;
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), (1,31,61,15,38,22,56,45)(2,30,62,14,39,21,49,44)(3,29,63,13,40,20,50,43)(4,28,64,12,33,19,51,42)(5,27,57,11,34,18,52,41)(6,26,58,10,35,17,53,48)(7,25,59,9,36,24,54,47)(8,32,60,16,37,23,55,46), (2,4)(3,7)(6,8)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,32)(18,27)(19,30)(20,25)(21,28)(22,31)(23,26)(24,29)(33,39)(35,37)(36,40)(49,51)(50,54)(53,55)(58,60)(59,63)(62,64) );
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)], [(1,31,61,15,38,22,56,45),(2,30,62,14,39,21,49,44),(3,29,63,13,40,20,50,43),(4,28,64,12,33,19,51,42),(5,27,57,11,34,18,52,41),(6,26,58,10,35,17,53,48),(7,25,59,9,36,24,54,47),(8,32,60,16,37,23,55,46)], [(2,4),(3,7),(6,8),(9,43),(10,46),(11,41),(12,44),(13,47),(14,42),(15,45),(16,48),(17,32),(18,27),(19,30),(20,25),(21,28),(22,31),(23,26),(24,29),(33,39),(35,37),(36,40),(49,51),(50,54),(53,55),(58,60),(59,63),(62,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8L | 8M | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | C8○D4 | C8⋊C22 | C8.C22 | C8.26D4 |
| kernel | C8⋊M4(2) | C8⋊C8 | D4⋊C8 | Q8⋊C8 | C8⋊1C8 | C8⋊6D4 | C4×SD16 | C8⋊4Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | C2×C8 | C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C8⋊M4(2) ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 15 |
| 0 | 0 | 14 | 11 | 1 | 2 |
| 0 | 0 | 0 | 2 | 0 | 11 |
| 0 | 0 | 16 | 15 | 3 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,16,0,0,6,11,2,15,0,0,0,1,0,3,0,0,15,2,11,6],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,1,16,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
C8⋊M4(2) in GAP, Magma, Sage, TeX
C_8\rtimes M_4(2)
% in TeX
G:=Group("C8:M4(2)"); // GroupNames label
G:=SmallGroup(128,324);
// by ID
G=gap.SmallGroup(128,324);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,723,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^5>;
// generators/relations