direct product, non-abelian, soluble
Aliases: C2×C8.A4, C8○D4⋊4C6, C4.8(C4×A4), C8.8(C2×A4), (C2×C8).2A4, C4.A4.4C4, C4○D4.3C12, (C2×Q8).6C12, Q8.4(C2×C12), C22.11(C4×A4), C4.14(C22×A4), C4.A4.18C22, (C2×SL2(𝔽3)).4C4, SL2(𝔽3).9(C2×C4), (C2×C8○D4)⋊C3, C2.10(C2×C4×A4), (C2×C4).20(C2×A4), (C2×C4○D4).6C6, C4○D4.11(C2×C6), (C2×C4.A4).10C2, SmallGroup(192,1012)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C2×C8.A4 |
Generators and relations for C2×C8.A4
G = < a,b,c,d,e | a2=b8=e3=1, c2=d2=b4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b4c, ece-1=b4cd, ede-1=c >
Subgroups: 197 in 76 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C2×C12, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C2×C24, C2×SL2(𝔽3), C4.A4, C2×C8○D4, C8.A4, C2×C4.A4, C2×C8.A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C22×A4, C8.A4, C2×C4×A4, C2×C8.A4
►On 64 points
Generators in S64
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(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 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)(17 49 21 53)(18 50 22 54)(19 51 23 55)(20 52 24 56)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)
(1 32 5 28)(2 25 6 29)(3 26 7 30)(4 27 8 31)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(9 63 26)(10 64 27)(11 57 28)(12 58 29)(13 59 30)(14 60 31)(15 61 32)(16 62 25)(33 45 53)(34 46 54)(35 47 55)(36 48 56)(37 41 49)(38 42 50)(39 43 51)(40 44 52)
G:=sub<Sym(64)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (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), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29)(17,49,21,53)(18,50,22,54)(19,51,23,55)(20,52,24,56)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (9,63,26)(10,64,27)(11,57,28)(12,58,29)(13,59,30)(14,60,31)(15,61,32)(16,62,25)(33,45,53)(34,46,54)(35,47,55)(36,48,56)(37,41,49)(38,42,50)(39,43,51)(40,44,52)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (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), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29)(17,49,21,53)(18,50,22,54)(19,51,23,55)(20,52,24,56)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (9,63,26)(10,64,27)(11,57,28)(12,58,29)(13,59,30)(14,60,31)(15,61,32)(16,62,25)(33,45,53)(34,46,54)(35,47,55)(36,48,56)(37,41,49)(38,42,50)(39,43,51)(40,44,52) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(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)], [(1,61,5,57),(2,62,6,58),(3,63,7,59),(4,64,8,60),(9,30,13,26),(10,31,14,27),(11,32,15,28),(12,25,16,29),(17,49,21,53),(18,50,22,54),(19,51,23,55),(20,52,24,56),(33,45,37,41),(34,46,38,42),(35,47,39,43),(36,48,40,44)], [(1,32,5,28),(2,25,6,29),(3,26,7,30),(4,27,8,31),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(9,63,26),(10,64,27),(11,57,28),(12,58,29),(13,59,30),(14,60,31),(15,61,32),(16,62,25),(33,45,53),(34,46,54),(35,47,55),(36,48,56),(37,41,49),(38,42,50),(39,43,51),(40,44,52)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 4 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | C8.A4 | A4 | C2×A4 | C2×A4 | C4×A4 | C4×A4 |
| kernel | C2×C8.A4 | C8.A4 | C2×C4.A4 | C2×C8○D4 | C2×SL2(𝔽3) | C4.A4 | C8○D4 | C2×C4○D4 | C2×Q8 | C4○D4 | C2 | C2×C8 | C8 | C2×C4 | C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 24 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C2×C8.A4 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 9 | 65 |
| 0 | 65 | 64 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 72 | 0 |
| 64 | 0 | 0 |
| 0 | 1 | 65 |
| 0 | 0 | 64 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,10,0,0,0,10],[1,0,0,0,9,65,0,65,64],[1,0,0,0,0,72,0,1,0],[64,0,0,0,1,0,0,65,64] >;
C2×C8.A4 in GAP, Magma, Sage, TeX
C_2\times C_8.A_4
% in TeX
G:=Group("C2xC8.A4"); // GroupNames label
G:=SmallGroup(192,1012);
// by ID
G=gap.SmallGroup(192,1012);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,92,248,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=e^3=1,c^2=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^4*c,e*c*e^-1=b^4*c*d,e*d*e^-1=c>;
// generators/relations