p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.391C23, C8⋊Q8⋊9C2, (C4×D8)⋊30C2, C8⋊2D4⋊7C2, C8⋊D4⋊9C2, C8⋊3D4⋊7C2, C8⋊8D4⋊15C2, C8⋊7D4⋊27C2, C4⋊C4.251D4, D8⋊C4⋊15C2, (C4×SD16)⋊18C2, C8.30(C4○D4), C2.26(D4○D8), C4.4D8⋊42C2, C22⋊C4.91D4, C23.88(C2×D4), C4⋊C4.118C23, (C4×C8).184C22, (C2×C8).279C23, (C2×C4).377C24, (C4×D4).97C22, C4⋊Q8.119C22, SD16⋊C4⋊22C2, (C4×Q8).94C22, C8○2M4(2)⋊20C2, C4.Q8.29C22, C2.40(D4○SD16), (C2×D4).131C23, (C2×D8).164C22, C4⋊D4.38C22, C4⋊1D4.66C22, (C2×Q8).119C23, C2.D8.221C22, C8⋊C4.134C22, C22⋊Q8.38C22, (C22×C8).279C22, (C2×SD16).25C22, C4.4D4.37C22, C22.637(C22×D4), C42.C2.23C22, D4⋊C4.149C22, C22.36C24⋊6C2, C22.34C24⋊4C2, (C22×C4).1057C23, Q8⋊C4.141C22, C42.78C22⋊30C2, C42⋊C2.334C22, (C2×M4(2)).287C22, C2.74(C22.26C24), C4.62(C2×C4○D4), (C2×C4).149(C2×D4), SmallGroup(128,1911)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 187 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×11], Q8 [×3], C23, C23 [×3], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×5], C2×Q8, C2×Q8, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8○2M4(2), C4×D8, C4×SD16, SD16⋊C4, D8⋊C4, C8⋊8D4, C8⋊7D4, C8⋊D4, C8⋊2D4, C4.4D8, C42.78C22, C8⋊3D4, C8⋊Q8, C22.34C24, C22.36C24, C42.391C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D4○D8, D4○SD16, C42.391C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=b, ab=ba, ac=ca, dad=ab2, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd=a2c, ece-1=b-1c, ede-1=b2d >
►On 64 points
(1 38 31 11)(2 39 32 12)(3 40 25 13)(4 33 26 14)(5 34 27 15)(6 35 28 16)(7 36 29 9)(8 37 30 10)(17 57 56 48)(18 58 49 41)(19 59 50 42)(20 60 51 43)(21 61 52 44)(22 62 53 45)(23 63 54 46)(24 64 55 47)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 54 5 50)(2 49 6 53)(3 52 7 56)(4 55 8 51)(9 57 13 61)(10 60 14 64)(11 63 15 59)(12 58 16 62)(17 25 21 29)(18 28 22 32)(19 31 23 27)(20 26 24 30)(33 47 37 43)(34 42 38 46)(35 45 39 41)(36 48 40 44)
(1 42)(2 47)(3 44)(4 41)(5 46)(6 43)(7 48)(8 45)(9 52)(10 49)(11 54)(12 51)(13 56)(14 53)(15 50)(16 55)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(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)
G:=sub<Sym(64)| (1,38,31,11)(2,39,32,12)(3,40,25,13)(4,33,26,14)(5,34,27,15)(6,35,28,16)(7,36,29,9)(8,37,30,10)(17,57,56,48)(18,58,49,41)(19,59,50,42)(20,60,51,43)(21,61,52,44)(22,62,53,45)(23,63,54,46)(24,64,55,47), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,54,5,50)(2,49,6,53)(3,52,7,56)(4,55,8,51)(9,57,13,61)(10,60,14,64)(11,63,15,59)(12,58,16,62)(17,25,21,29)(18,28,22,32)(19,31,23,27)(20,26,24,30)(33,47,37,43)(34,42,38,46)(35,45,39,41)(36,48,40,44), (1,42)(2,47)(3,44)(4,41)(5,46)(6,43)(7,48)(8,45)(9,52)(10,49)(11,54)(12,51)(13,56)(14,53)(15,50)(16,55)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(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)>;
G:=Group( (1,38,31,11)(2,39,32,12)(3,40,25,13)(4,33,26,14)(5,34,27,15)(6,35,28,16)(7,36,29,9)(8,37,30,10)(17,57,56,48)(18,58,49,41)(19,59,50,42)(20,60,51,43)(21,61,52,44)(22,62,53,45)(23,63,54,46)(24,64,55,47), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,54,5,50)(2,49,6,53)(3,52,7,56)(4,55,8,51)(9,57,13,61)(10,60,14,64)(11,63,15,59)(12,58,16,62)(17,25,21,29)(18,28,22,32)(19,31,23,27)(20,26,24,30)(33,47,37,43)(34,42,38,46)(35,45,39,41)(36,48,40,44), (1,42)(2,47)(3,44)(4,41)(5,46)(6,43)(7,48)(8,45)(9,52)(10,49)(11,54)(12,51)(13,56)(14,53)(15,50)(16,55)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(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) );
G=PermutationGroup([(1,38,31,11),(2,39,32,12),(3,40,25,13),(4,33,26,14),(5,34,27,15),(6,35,28,16),(7,36,29,9),(8,37,30,10),(17,57,56,48),(18,58,49,41),(19,59,50,42),(20,60,51,43),(21,61,52,44),(22,62,53,45),(23,63,54,46),(24,64,55,47)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,54,5,50),(2,49,6,53),(3,52,7,56),(4,55,8,51),(9,57,13,61),(10,60,14,64),(11,63,15,59),(12,58,16,62),(17,25,21,29),(18,28,22,32),(19,31,23,27),(20,26,24,30),(33,47,37,43),(34,42,38,46),(35,45,39,41),(36,48,40,44)], [(1,42),(2,47),(3,44),(4,41),(5,46),(6,43),(7,48),(8,45),(9,52),(10,49),(11,54),(12,51),(13,56),(14,53),(15,50),(16,55),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,61),(26,58),(27,63),(28,60),(29,57),(30,62),(31,59),(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)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 9 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 12 |
| 0 | 0 | 0 | 5 | 12 | 0 |
| 0 | 0 | 0 | 12 | 12 | 0 |
| 0 | 0 | 12 | 0 | 0 | 5 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 | 0 |
| 0 | 0 | 5 | 0 | 0 | 12 |
| 0 | 0 | 5 | 0 | 0 | 5 |
| 0 | 0 | 0 | 5 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,12,0,0,12,0,0,0,5,12,0,0,0,0,12,12,0,0,0,12,0,0,5],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,5,5,0,0,0,12,0,0,5,0,0,12,0,0,12,0,0,0,12,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,12,12,0,0,0,0,5,12,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○D8 | D4○SD16 |
| kernel | C42.391C23 | C8○2M4(2) | C4×D8 | C4×SD16 | SD16⋊C4 | D8⋊C4 | C8⋊8D4 | C8⋊7D4 | C8⋊D4 | C8⋊2D4 | C4.4D8 | C42.78C22 | C8⋊3D4 | C8⋊Q8 | C22.34C24 | C22.36C24 | C22⋊C4 | C4⋊C4 | C8 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{391}C_2^3 % in TeX
G:=Group("C4^2.391C2^3"); // GroupNames label
G:=SmallGroup(128,1911);
// by ID
G=gap.SmallGroup(128,1911);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,184,1018,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b^2,e^2=b,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d=a^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d>;
// generators/relations