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 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], 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 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×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 54 23 33 9 45)(2 30 62 53 24 40 10 44)(3 29 63 52 17 39 11 43)(4 28 64 51 18 38 12 42)(5 27 57 50 19 37 13 41)(6 26 58 49 20 36 14 48)(7 25 59 56 21 35 15 47)(8 32 60 55 22 34 16 46)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)(41 50)(42 53)(43 56)(44 51)(45 54)(46 49)(47 52)(48 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), (1,31,61,54,23,33,9,45)(2,30,62,53,24,40,10,44)(3,29,63,52,17,39,11,43)(4,28,64,51,18,38,12,42)(5,27,57,50,19,37,13,41)(6,26,58,49,20,36,14,48)(7,25,59,56,21,35,15,47)(8,32,60,55,22,34,16,46), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)(41,50)(42,53)(43,56)(44,51)(45,54)(46,49)(47,52)(48,55)(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), (1,31,61,54,23,33,9,45)(2,30,62,53,24,40,10,44)(3,29,63,52,17,39,11,43)(4,28,64,51,18,38,12,42)(5,27,57,50,19,37,13,41)(6,26,58,49,20,36,14,48)(7,25,59,56,21,35,15,47)(8,32,60,55,22,34,16,46), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)(41,50)(42,53)(43,56)(44,51)(45,54)(46,49)(47,52)(48,55)(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)], [(1,31,61,54,23,33,9,45),(2,30,62,53,24,40,10,44),(3,29,63,52,17,39,11,43),(4,28,64,51,18,38,12,42),(5,27,57,50,19,37,13,41),(6,26,58,49,20,36,14,48),(7,25,59,56,21,35,15,47),(8,32,60,55,22,34,16,46)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,21),(18,24),(20,22),(25,39),(26,34),(27,37),(28,40),(29,35),(30,38),(31,33),(32,36),(41,50),(42,53),(43,56),(44,51),(45,54),(46,49),(47,52),(48,55),(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