p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.678C23, (C4×D4).25C4, (C4×Q8).24C4, C4.48(C8○D4), C4⋊C8.231C22, (C4×C8).327C22, (C2×C8).400C23, (C2×C4).644C24, C42.207(C2×C4), C8○2M4(2)⋊29C2, C4⋊M4(2)⋊33C2, C42.6C4⋊46C2, C8⋊C4.154C22, C42.12C4⋊48C2, C4.44(C42⋊C2), C2.13(Q8○M4(2)), C22⋊C8.139C22, (C2×C42).757C22, (C22×C4).914C23, C23.102(C22×C4), (C22×C8).432C22, C22.172(C23×C4), C42.7C22⋊22C2, C42⋊C2.350C22, C22.18(C42⋊C2), (C2×M4(2)).346C22, (C2×C4⋊C8)⋊46C2, C2.13(C2×C8○D4), C4⋊C4.220(C2×C4), C4⋊C8○(C42⋊C2), (C4×C4○D4).13C2, C4.295(C2×C4○D4), (C2×D4).229(C2×C4), C22⋊C4.70(C2×C4), (C2×Q8).207(C2×C4), (C2×C4).829(C4○D4), (C2×C4).260(C22×C4), (C22×C4).338(C2×C4), C2.44(C2×C42⋊C2), (C22×C8)⋊C2.19C2, (C2×C4○D4).282C22, SmallGroup(128,1657)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 196 conjugacy classes, 134 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×10], C2×C4 [×13], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×2], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C8○2M4(2) [×2], (C22×C8)⋊C2 [×2], C2×C4⋊C8, C4⋊M4(2), C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C4×C4○D4, C42.678C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.678C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >
►On 64 points
(1 63 55 41)(2 42 56 64)(3 57 49 43)(4 44 50 58)(5 59 51 45)(6 46 52 60)(7 61 53 47)(8 48 54 62)(9 21 28 36)(10 37 29 22)(11 23 30 38)(12 39 31 24)(13 17 32 40)(14 33 25 18)(15 19 26 34)(16 35 27 20)
(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 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 56)(4 50)(6 52)(8 54)(9 32)(10 14)(11 26)(12 16)(13 28)(15 30)(17 36)(18 22)(19 38)(20 24)(21 40)(23 34)(25 29)(27 31)(33 37)(35 39)(42 64)(44 58)(46 60)(48 62)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
G:=sub<Sym(64)| (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,21,28,36)(10,37,29,22)(11,23,30,38)(12,39,31,24)(13,17,32,40)(14,33,25,18)(15,19,26,34)(16,35,27,20), (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), (1,2,3,4,5,6,7,8)(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), (2,56)(4,50)(6,52)(8,54)(9,32)(10,14)(11,26)(12,16)(13,28)(15,30)(17,36)(18,22)(19,38)(20,24)(21,40)(23,34)(25,29)(27,31)(33,37)(35,39)(42,64)(44,58)(46,60)(48,62), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)>;
G:=Group( (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,21,28,36)(10,37,29,22)(11,23,30,38)(12,39,31,24)(13,17,32,40)(14,33,25,18)(15,19,26,34)(16,35,27,20), (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), (1,2,3,4,5,6,7,8)(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), (2,56)(4,50)(6,52)(8,54)(9,32)(10,14)(11,26)(12,16)(13,28)(15,30)(17,36)(18,22)(19,38)(20,24)(21,40)(23,34)(25,29)(27,31)(33,37)(35,39)(42,64)(44,58)(46,60)(48,62), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64) );
G=PermutationGroup([(1,63,55,41),(2,42,56,64),(3,57,49,43),(4,44,50,58),(5,59,51,45),(6,46,52,60),(7,61,53,47),(8,48,54,62),(9,21,28,36),(10,37,29,22),(11,23,30,38),(12,39,31,24),(13,17,32,40),(14,33,25,18),(15,19,26,34),(16,35,27,20)], [(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)], [(1,2,3,4,5,6,7,8),(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)], [(2,56),(4,50),(6,52),(8,54),(9,32),(10,14),(11,26),(12,16),(13,28),(15,30),(17,36),(18,22),(19,38),(20,24),(21,40),(23,34),(25,29),(27,31),(33,37),(35,39),(42,64),(44,58),(46,60),(48,62)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.678C23 | C8○2M4(2) | (C22×C8)⋊C2 | C2×C4⋊C8 | C4⋊M4(2) | C42.12C4 | C42.6C4 | C42.7C22 | C4×C4○D4 | C4×D4 | C4×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 1 | 12 | 4 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{678}C_2^3 % in TeX
G:=Group("C4^2.678C2^3"); // GroupNames label
G:=SmallGroup(128,1657);
// by ID
G=gap.SmallGroup(128,1657);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations