direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊4D4, C42.357D4, C42.711C23, C4⋊1(C2×D8), (C2×C4)⋊7D8, (C2×C8)⋊30D4, C8⋊10(C2×D4), (C4×C8)⋊73C22, C4.2(C22×D4), (C22×D8)⋊11C2, (C2×D8)⋊44C22, C2.11(C22×D8), C22.73(C2×D8), C4.12(C4⋊1D4), C4⋊1D4⋊36C22, (C2×C4).342C24, (C2×C8).558C23, (C22×C4).612D4, C23.877(C2×D4), (C2×D4).109C23, C22.48(C4⋊1D4), (C22×C8).535C22, C22.602(C22×D4), (C2×C42).1126C22, (C22×C4).1557C23, (C22×D4).372C22, (C2×C4×C8)⋊25C2, (C2×C4⋊1D4)⋊17C2, (C2×C4).852(C2×D4), C2.21(C2×C4⋊1D4), SmallGroup(128,1876)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 980 in 380 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×40], C8 [×8], C2×C4 [×2], C2×C4 [×16], D4 [×56], C23, C23 [×32], C42 [×4], C2×C8 [×12], D8 [×32], C22×C4, C22×C4 [×2], C2×D4 [×8], C2×D4 [×52], C24 [×4], C4×C8 [×4], C2×C42, C4⋊1D4 [×8], C4⋊1D4 [×4], C22×C8 [×2], C2×D8 [×16], C2×D8 [×16], C22×D4 [×4], C22×D4 [×4], C2×C4×C8, C8⋊4D4 [×8], C2×C4⋊1D4 [×2], C22×D8 [×4], C2×C8⋊4D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D8 [×8], C2×D4 [×18], C24, C4⋊1D4 [×4], C2×D8 [×12], C22×D4 [×3], C8⋊4D4 [×4], C2×C4⋊1D4, C22×D8 [×2], C2×C8⋊4D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
►On 64 points
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(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 32 37 59)(2 25 38 60)(3 26 39 61)(4 27 40 62)(5 28 33 63)(6 29 34 64)(7 30 35 57)(8 31 36 58)(9 43 52 20)(10 44 53 21)(11 45 54 22)(12 46 55 23)(13 47 56 24)(14 48 49 17)(15 41 50 18)(16 42 51 19)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 56)(41 60)(42 59)(43 58)(44 57)(45 64)(46 63)(47 62)(48 61)
G:=sub<Sym(64)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,32,37,59)(2,25,38,60)(3,26,39,61)(4,27,40,62)(5,28,33,63)(6,29,34,64)(7,30,35,57)(8,31,36,58)(9,43,52,20)(10,44,53,21)(11,45,54,22)(12,46,55,23)(13,47,56,24)(14,48,49,17)(15,41,50,18)(16,42,51,19), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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), (1,32,37,59)(2,25,38,60)(3,26,39,61)(4,27,40,62)(5,28,33,63)(6,29,34,64)(7,30,35,57)(8,31,36,58)(9,43,52,20)(10,44,53,21)(11,45,54,22)(12,46,55,23)(13,47,56,24)(14,48,49,17)(15,41,50,18)(16,42,51,19), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,60)(42,59)(43,58)(44,57)(45,64)(46,63)(47,62)(48,61) );
G=PermutationGroup([(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(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)], [(1,32,37,59),(2,25,38,60),(3,26,39,61),(4,27,40,62),(5,28,33,63),(6,29,34,64),(7,30,35,57),(8,31,36,58),(9,43,52,20),(10,44,53,21),(11,45,54,22),(12,46,55,23),(13,47,56,24),(14,48,49,17),(15,41,50,18),(16,42,51,19)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,26),(18,25),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,56),(41,60),(42,59),(43,58),(44,57),(45,64),(46,63),(47,62),(48,61)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 14 | 14 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,3,0,0,0,11,11,0,0,0,0,0,14,14,0,0,0,3,14],[16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 |
| kernel | C2×C8⋊4D4 | C2×C4×C8 | C8⋊4D4 | C2×C4⋊1D4 | C22×D8 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 8 | 2 | 4 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_4D_4
% in TeX
G:=Group("C2xC8:4D4"); // GroupNames label
G:=SmallGroup(128,1876);
// by ID
G=gap.SmallGroup(128,1876);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,184,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations