direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C4○D16, D16⋊7C22, C8.12C24, Q32⋊6C22, D8.2C23, C23.29D8, C16.13C23, SD32⋊6C22, Q16.2C23, C4○(C2×D16), (C2×C4)○D16, C4○(C2×Q32), (C2×C4)○Q32, C4○(C2×SD32), (C2×C4)○SD32, C4○(C4○D16), (C2×C4).97D8, C4.95(C2×D8), C8.57(C2×D4), (C2×D16)⋊14C2, (C2×Q32)⋊14C2, (C2×C8).272D4, C4○D8⋊5C22, C22.3(C2×D8), (C2×C16)⋊19C22, (C22×C16)⋊10C2, (C2×SD32)⋊18C2, C4.18(C22×D4), C2.27(C22×D8), (C2×C8).585C23, (C22×C4).624D4, (C2×D8).151C22, (C22×C8).560C22, (C2×Q16).147C22, (C2×C4)○(C2×D16), (C2×C4)○(C2×Q32), (C2×C4)○(C2×SD32), (C2×C4○D8)⋊15C2, (C2×C4).875(C2×D4), SmallGroup(128,2143)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 404 in 184 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×2], SD16 [×8], Q16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C2×C16 [×2], C2×C16 [×4], D16 [×4], SD32 [×8], Q32 [×4], C22×C8, C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4○D8 [×8], C4○D8 [×4], C2×C4○D4 [×2], C22×C16, C2×D16, C2×SD32 [×2], C2×Q32, C4○D16 [×8], C2×C4○D8 [×2], C2×C4○D16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C4○D16 [×2], C22×D8, C2×C4○D16
Generators and relations
G = < a,b,c,d | a2=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >
►On 64 points
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 49)
(1 51 9 59)(2 52 10 60)(3 53 11 61)(4 54 12 62)(5 55 13 63)(6 56 14 64)(7 57 15 49)(8 58 16 50)(17 46 25 38)(18 47 26 39)(19 48 27 40)(20 33 28 41)(21 34 29 42)(22 35 30 43)(23 36 31 44)(24 37 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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 25)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)(33 35)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 53)(50 52)(54 64)(55 63)(56 62)(57 61)(58 60)
G:=sub<Sym(64)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,49), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,46,25,38)(18,47,26,39)(19,48,27,40)(20,33,28,41)(21,34,29,42)(22,35,30,43)(23,36,31,44)(24,37,32,45), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,25)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,53)(50,52)(54,64)(55,63)(56,62)(57,61)(58,60)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,17)(14,18)(15,19)(16,20)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,49), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,46,25,38)(18,47,26,39)(19,48,27,40)(20,33,28,41)(21,34,29,42)(22,35,30,43)(23,36,31,44)(24,37,32,45), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,25)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,35)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(49,53)(50,52)(54,64)(55,63)(56,62)(57,61)(58,60) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,17),(14,18),(15,19),(16,20),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,49)], [(1,51,9,59),(2,52,10,60),(3,53,11,61),(4,54,12,62),(5,55,13,63),(6,56,14,64),(7,57,15,49),(8,58,16,50),(17,46,25,38),(18,47,26,39),(19,48,27,40),(20,33,28,41),(21,34,29,42),(22,35,30,43),(23,36,31,44),(24,37,32,45)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,25),(18,24),(19,23),(20,22),(26,32),(27,31),(28,30),(33,35),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(49,53),(50,52),(54,64),(55,63),(56,62),(57,61),(58,60)])
Matrix representation ►G ⊆ GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 16 | 0 | 0 |
| 0 | 7 | 12 |
| 0 | 11 | 2 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,4,0,0,0,4],[16,0,0,0,7,11,0,12,2],[16,0,0,0,1,1,0,0,16] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | C4○D16 |
| kernel | C2×C4○D16 | C22×C16 | C2×D16 | C2×SD32 | C2×Q32 | C4○D16 | C2×C4○D8 | C2×C8 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 1 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\circ D_{16} % in TeX
G:=Group("C2xC4oD16"); // GroupNames label
G:=SmallGroup(128,2143);
// by ID
G=gap.SmallGroup(128,2143);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,352,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations