direct product, non-abelian, soluble
Aliases: C2×C8.A4, C8○D4⋊4C6, C8.8(C2×A4), C4.8(C4×A4), (C2×C8).2A4, C4.A4.4C4, C4○D4.3C12, (C2×Q8).6C12, Q8.4(C2×C12), C4.14(C22×A4), C22.11(C4×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 |
Subgroups: 197 in 76 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×4], Q8, Q8, C23, C12 [×2], C2×C6, C2×C8, C2×C8 [×5], M4(2) [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×2], C4○D4 [×2], C24 [×2], SL2(𝔽3), C2×C12, C22×C8, C2×M4(2), C8○D4 [×2], C8○D4 [×2], C2×C4○D4, C2×C24, C2×SL2(𝔽3), C4.A4 [×2], C2×C8○D4, C8.A4 [×2], C2×C4.A4, C2×C8.A4
Quotients:
C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], A4, C2×C6, C2×C12, C2×A4 [×3], C4×A4 [×2], C22×A4, C8.A4 [×2], C2×C4×A4, C2×C8.A4
Generators and relations
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 >
►On 64 points
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(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 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(17 53 21 49)(18 54 22 50)(19 55 23 51)(20 56 24 52)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 40 21 36)(18 33 22 37)(19 34 23 38)(20 35 24 39)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(9 63 32)(10 64 25)(11 57 26)(12 58 27)(13 59 28)(14 60 29)(15 61 30)(16 62 31)(33 46 54)(34 47 55)(35 48 56)(36 41 49)(37 42 50)(38 43 51)(39 44 52)(40 45 53)
G:=sub<Sym(64)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(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)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,53,21,49)(18,54,22,50)(19,55,23,51)(20,56,24,52)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,40,21,36)(18,33,22,37)(19,34,23,38)(20,35,24,39)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (9,63,32)(10,64,25)(11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,61,30)(16,62,31)(33,46,54)(34,47,55)(35,48,56)(36,41,49)(37,42,50)(38,43,51)(39,44,52)(40,45,53)>;
G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(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)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,53,21,49)(18,54,22,50)(19,55,23,51)(20,56,24,52)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,40,21,36)(18,33,22,37)(19,34,23,38)(20,35,24,39)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (9,63,32)(10,64,25)(11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,61,30)(16,62,31)(33,46,54)(34,47,55)(35,48,56)(36,41,49)(37,42,50)(38,43,51)(39,44,52)(40,45,53) );
G=PermutationGroup([(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(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),(9,28,13,32),(10,29,14,25),(11,30,15,26),(12,31,16,27),(17,53,21,49),(18,54,22,50),(19,55,23,51),(20,56,24,52),(33,46,37,42),(34,47,38,43),(35,48,39,44),(36,41,40,45)], [(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,40,21,36),(18,33,22,37),(19,34,23,38),(20,35,24,39),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(9,63,32),(10,64,25),(11,57,26),(12,58,27),(13,59,28),(14,60,29),(15,61,30),(16,62,31),(33,46,54),(34,47,55),(35,48,56),(36,41,49),(37,42,50),(38,43,51),(39,44,52),(40,45,53)])
Matrix representation ►G ⊆ 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] >;
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 |
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