direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×M5(2), C16⋊4C23, C24.6C8, C8.23C24, (C2×C4)○M5(2), C4○(C2×M5(2)), C8○2(C2×M5(2)), (C2×C8)○2M5(2), (C22×C16)⋊14C2, (C2×C16)⋊21C22, (C22×C8).50C4, C23.41(C2×C8), (C23×C8).26C2, C2.10(C23×C8), C4.62(C23×C4), C8.63(C22×C4), (C22×C4).18C8, C4.39(C22×C8), (C23×C4).43C4, (C2×C8).616C23, C22.16(C22×C8), (C22×C8).586C22, (C2×C4).90(C2×C8), (C2×C8)○(C2×M5(2)), (C2×C4)○(C2×M5(2)), (C2×C8).255(C2×C4), (C2×C4).627(C22×C4), (C22×C4).507(C2×C4), SmallGroup(128,2137)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×M5(2)
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >
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)
►On 64 points
(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 49)(46 50)(47 51)(48 52)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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 40)(2 33)(3 42)(4 35)(5 44)(6 37)(7 46)(8 39)(9 48)(10 41)(11 34)(12 43)(13 36)(14 45)(15 38)(16 47)(17 57)(18 50)(19 59)(20 52)(21 61)(22 54)(23 63)(24 56)(25 49)(26 58)(27 51)(28 60)(29 53)(30 62)(31 55)(32 64)
G:=sub<Sym(64)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,40)(2,33)(3,42)(4,35)(5,44)(6,37)(7,46)(8,39)(9,48)(10,41)(11,34)(12,43)(13,36)(14,45)(15,38)(16,47)(17,57)(18,50)(19,59)(20,52)(21,61)(22,54)(23,63)(24,56)(25,49)(26,58)(27,51)(28,60)(29,53)(30,62)(31,55)(32,64)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,40)(2,33)(3,42)(4,35)(5,44)(6,37)(7,46)(8,39)(9,48)(10,41)(11,34)(12,43)(13,36)(14,45)(15,38)(16,47)(17,57)(18,50)(19,59)(20,52)(21,61)(22,54)(23,63)(24,56)(25,49)(26,58)(27,51)(28,60)(29,53)(30,62)(31,55)(32,64) );
G=PermutationGroup([(1,28),(2,29),(3,30),(4,31),(5,32),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,49),(46,50),(47,51),(48,52)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(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)], [(1,40),(2,33),(3,42),(4,35),(5,44),(6,37),(7,46),(8,39),(9,48),(10,41),(11,34),(12,43),(13,36),(14,45),(15,38),(16,47),(17,57),(18,50),(19,59),(20,52),(21,61),(22,54),(23,63),(24,56),(25,49),(26,58),(27,51),(28,60),(29,53),(30,62),(31,55),(32,64)])
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 |
Matrix representation of C22×M5(2) ►in 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] >;
C22×M5(2) 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