direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4×M4(2), C42○M4(2), C23.37C42, C42.748C23, C8⋊7(C22×C4), C4○(C4×M4(2)), (C4×C8)⋊78C22, (C23×C4).33C4, C4.57(C23×C4), (C2×C4).76C42, (C2×C42).52C4, C4.30(C2×C42), C42○(C4×M4(2)), C42○2(C8⋊C4), C8⋊C4⋊67C22, C42.331(C2×C4), C24.123(C2×C4), (C2×C8).610C23, (C2×C4).621C24, C42○2(C2×M4(2)), C2.13(C22×C42), C22.16(C2×C42), (C22×C42).29C2, C22.32(C23×C4), C2.2(C22×M4(2)), C42○(C22×M4(2)), (C22×C8).582C22, C23.287(C22×C4), (C23×C4).685C22, C22.58(C2×M4(2)), (C22×C4).1647C23, (C2×C42).1100C22, (C22×M4(2)).32C2, (C2×M4(2)).378C22, (C2×C4×C8)⋊39C2, C4○2(C2×C8⋊C4), (C2×C8)⋊32(C2×C4), C8⋊C4○(C2×C42), (C2×C8⋊C4)⋊40C2, (C2×C4)○(C4×M4(2)), (C2×C4)○3(C8⋊C4), C42○2(C2×C8⋊C4), (C2×C42)○(C2×M4(2)), (C2×C4).623(C22×C4), (C22×C4).455(C2×C4), (C2×C42)○(C22×M4(2)), (C2×C4)○2(C2×C8⋊C4), SmallGroup(128,1603)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 328 conjugacy classes, 276 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×16], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×24], M4(2) [×32], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C4×C8 [×8], C8⋊C4 [×8], C2×C42 [×2], C2×C42 [×10], C22×C8 [×4], C2×M4(2) [×24], C23×C4, C23×C4 [×2], C2×C4×C8 [×2], C2×C8⋊C4 [×2], C4×M4(2) [×8], C22×C42, C22×M4(2) [×2], C2×C4×M4(2)
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], M4(2) [×8], C22×C4 [×42], C24, C2×C42 [×12], C2×M4(2) [×12], C23×C4 [×3], C4×M4(2) [×4], C22×C42, C22×M4(2) [×2], C2×C4×M4(2)
Generators and relations
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
►On 64 points
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 11 61 32)(2 12 62 25)(3 13 63 26)(4 14 64 27)(5 15 57 28)(6 16 58 29)(7 9 59 30)(8 10 60 31)(17 34 50 43)(18 35 51 44)(19 36 52 45)(20 37 53 46)(21 38 54 47)(22 39 55 48)(23 40 56 41)(24 33 49 42)
(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)(2 58)(3 63)(4 60)(5 57)(6 62)(7 59)(8 64)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)
G:=sub<Sym(64)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,11,61,32)(2,12,62,25)(3,13,63,26)(4,14,64,27)(5,15,57,28)(6,16,58,29)(7,9,59,30)(8,10,60,31)(17,34,50,43)(18,35,51,44)(19,36,52,45)(20,37,53,46)(21,38,54,47)(22,39,55,48)(23,40,56,41)(24,33,49,42), (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)(2,58)(3,63)(4,60)(5,57)(6,62)(7,59)(8,64)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,11,61,32)(2,12,62,25)(3,13,63,26)(4,14,64,27)(5,15,57,28)(6,16,58,29)(7,9,59,30)(8,10,60,31)(17,34,50,43)(18,35,51,44)(19,36,52,45)(20,37,53,46)(21,38,54,47)(22,39,55,48)(23,40,56,41)(24,33,49,42), (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)(2,58)(3,63)(4,60)(5,57)(6,62)(7,59)(8,64)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)], [(1,11,61,32),(2,12,62,25),(3,13,63,26),(4,14,64,27),(5,15,57,28),(6,16,58,29),(7,9,59,30),(8,10,60,31),(17,34,50,43),(18,35,51,44),(19,36,52,45),(20,37,53,46),(21,38,54,47),(22,39,55,48),(23,40,56,41),(24,33,49,42)], [(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),(2,58),(3,63),(4,60),(5,57),(6,62),(7,59),(8,64),(9,30),(10,27),(11,32),(12,29),(13,26),(14,31),(15,28),(16,25),(17,50),(18,55),(19,52),(20,49),(21,54),(22,51),(23,56),(24,53),(33,46),(34,43),(35,48),(36,45),(37,42),(38,47),(39,44),(40,41)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 9 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,1,0,0,0,0,13,9,0,0,11,4],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,16,16] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4AJ | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) |
| kernel | C2×C4×M4(2) | C2×C4×C8 | C2×C8⋊C4 | C4×M4(2) | C22×C42 | C22×M4(2) | C2×C42 | C2×M4(2) | C23×C4 | C2×C4 |
| # reps | 1 | 2 | 2 | 8 | 1 | 2 | 12 | 32 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times M_{4(2)} % in TeX
G:=Group("C2xC4xM4(2)"); // GroupNames label
G:=SmallGroup(128,1603);
// by ID
G=gap.SmallGroup(128,1603);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,1430,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations