direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×M5(2), C16⋊4C23, C24.6C8, C8.23C24, C4○(C2×M5(2)), (C2×C4)○M5(2), C8○2(C2×M5(2)), (C2×C8)○2M5(2), (C22×C16)⋊14C2, (C2×C16)⋊21C22, (C22×C4).18C8, C8.63(C22×C4), (C23×C4).43C4, C4.39(C22×C8), C4.62(C23×C4), (C22×C8).50C4, C2.10(C23×C8), (C23×C8).26C2, C23.41(C2×C8), (C2×C8).616C23, C22.16(C22×C8), (C22×C8).586C22, (C2×C4).90(C2×C8), (C2×C4)○(C2×M5(2)), (C2×C8)○(C2×M5(2)), (C2×C8).255(C2×C4), (C22×C4).507(C2×C4), (C2×C4).627(C22×C4), SmallGroup(128,2137)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 220 in 200 conjugacy classes, 180 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×7], C22 [×11], C22 [×12], C8, C8 [×7], C2×C4 [×28], C23, C23 [×6], C23 [×4], C16 [×8], C2×C8 [×28], C22×C4 [×2], C22×C4 [×12], C24, C2×C16 [×12], M5(2) [×16], C22×C8 [×2], C22×C8 [×12], C23×C4, C22×C16 [×2], C2×M5(2) [×12], C23×C8, C22×M5(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, M5(2) [×4], C22×C8 [×14], C23×C4, C2×M5(2) [×6], C23×C8, C22×M5(2)
Generators and relations
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >
►On 64 points
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 46)(18 47)(19 48)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 49)(2 58)(3 51)(4 60)(5 53)(6 62)(7 55)(8 64)(9 57)(10 50)(11 59)(12 52)(13 61)(14 54)(15 63)(16 56)(17 38)(18 47)(19 40)(20 33)(21 42)(22 35)(23 44)(24 37)(25 46)(26 39)(27 48)(28 41)(29 34)(30 43)(31 36)(32 45)
G:=sub<Sym(64)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,49)(2,58)(3,51)(4,60)(5,53)(6,62)(7,55)(8,64)(9,57)(10,50)(11,59)(12,52)(13,61)(14,54)(15,63)(16,56)(17,38)(18,47)(19,40)(20,33)(21,42)(22,35)(23,44)(24,37)(25,46)(26,39)(27,48)(28,41)(29,34)(30,43)(31,36)(32,45)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,46)(18,47)(19,48)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,49)(2,58)(3,51)(4,60)(5,53)(6,62)(7,55)(8,64)(9,57)(10,50)(11,59)(12,52)(13,61)(14,54)(15,63)(16,56)(17,38)(18,47)(19,40)(20,33)(21,42)(22,35)(23,44)(24,37)(25,46)(26,39)(27,48)(28,41)(29,34)(30,43)(31,36)(32,45) );
G=PermutationGroup([(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,49),(29,50),(30,51),(31,52),(32,53)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,46),(18,47),(19,48),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45)], [(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)], [(1,49),(2,58),(3,51),(4,60),(5,53),(6,62),(7,55),(8,64),(9,57),(10,50),(11,59),(12,52),(13,61),(14,54),(15,63),(16,56),(17,38),(18,47),(19,40),(20,33),(21,42),(22,35),(23,44),(24,37),(25,46),(26,39),(27,48),(28,41),(29,34),(30,43),(31,36),(32,45)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,8,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) |
| kernel | C22×M5(2) | C22×C16 | C2×M5(2) | C23×C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 28 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times M_{5(2)} % in TeX
G:=Group("C2^2xM5(2)"); // GroupNames label
G:=SmallGroup(128,2137);
// by ID
G=gap.SmallGroup(128,2137);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,925,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=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^9>;
// generators/relations