direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×C4○D8, D8⋊7C23, C4.4C25, C8.20C24, Q16⋊7C23, D4.2C24, Q8.2C24, SD16⋊6C23, C24.145D4, D8○(C22×C4), C4○(C22×D8), Q16○(C22×C4), C4○(C22×Q16), (C2×C8)⋊15C23, (C23×C8)⋊11C2, C4○D4⋊4C23, C4○(C22×SD16), SD16○(C22×C4), (C22×D8)⋊24C2, (C2×D8)⋊59C22, C4.30(C22×D4), C2.39(D4×C23), (C2×C4).610C24, (C22×C8)⋊67C22, (C2×Q16)⋊63C22, (C22×Q16)⋊24C2, (C22×C4).630D4, C23.410(C2×D4), C22.5(C22×D4), (C22×SD16)⋊30C2, (C2×SD16)⋊82C22, (C2×D4).490C23, (C2×Q8).474C23, (C23×C4).714C22, (C22×C4).1592C23, (C22×D4).603C22, (C22×Q8).504C22, C4○(C2×C4○D8), (C2×C4)○2(C2×D8), (C2×C4)○(C4○D8), (C2×C4)○2(C2×Q16), (C22×C4)○(C2×D8), (C2×C4)○(C22×D8), (C2×C4)○2(C2×SD16), (C22×C4)○(C2×Q16), (C2×C4)○(C22×Q16), (C2×C4).883(C2×D4), (C22×C4)○(C2×SD16), (C2×C4)○(C22×SD16), (C22×C4)○(C22×D8), (C2×C4○D4)⋊76C22, (C22×C4○D4)⋊25C2, (C22×C4)○(C22×Q16), (C22×C4)○(C22×SD16), (C2×C4)○(C2×C4○D8), SmallGroup(128,2309)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1148 in 752 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×12], C4, C4 [×7], C4 [×8], C22 [×11], C22 [×44], C8 [×8], C2×C4 [×28], C2×C4 [×44], D4 [×8], D4 [×44], Q8 [×8], Q8 [×12], C23, C23 [×6], C23 [×24], C2×C8 [×28], D8 [×16], SD16 [×32], Q16 [×16], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×12], C2×D4 [×30], C2×Q8 [×12], C2×Q8 [×6], C4○D4 [×32], C4○D4 [×48], C24, C24 [×2], C22×C8 [×2], C22×C8 [×12], C2×D8 [×12], C2×SD16 [×24], C2×Q16 [×12], C4○D8 [×64], C23×C4, C23×C4 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×24], C2×C4○D4 [×12], C23×C8, C22×D8, C22×SD16 [×2], C22×Q16, C2×C4○D8 [×24], C22×C4○D4 [×2], C22×C4○D8
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C4○D8 [×4], C22×D4 [×14], C25, C2×C4○D8 [×6], D4×C23, C22×C4○D8
Generators and relations
G = < a,b,c,d,e | a2=b2=c4=e2=1, d4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
►On 64 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 33 29 37)(26 34 30 38)(27 35 31 39)(28 36 32 40)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(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 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 64)(25 51)(26 50)(27 49)(28 56)(29 55)(30 54)(31 53)(32 52)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)
G:=sub<Sym(64)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,64)(25,51)(26,50)(27,49)(28,56)(29,55)(30,54)(31,53)(32,52)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,64)(25,51)(26,50)(27,49)(28,56)(29,55)(30,54)(31,53)(32,52)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48) );
G=PermutationGroup([(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,56),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(25,33,29,37),(26,34,30,38),(27,35,31,39),(28,36,32,40),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(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)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,64),(25,51),(26,50),(27,49),(28,56),(29,55),(30,54),(31,53),(32,52),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,48)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[16,0,0,0,0,1,0,0,0,0,3,3,0,0,14,3],[16,0,0,0,0,16,0,0,0,0,3,14,0,0,14,14] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 |
| kernel | C22×C4○D8 | C23×C8 | C22×D8 | C22×SD16 | C22×Q16 | C2×C4○D8 | C22×C4○D4 | C22×C4 | C24 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 24 | 2 | 7 | 1 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_8
% in TeX
G:=Group("C2^2xC4oD8"); // GroupNames label
G:=SmallGroup(128,2309);
// by ID
G=gap.SmallGroup(128,2309);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,352,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^4=c^2,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,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations