p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.266C23, (C2×C8)⋊27D4, C4.41(C4×D4), C4⋊Q8.28C4, C8⋊9D4⋊31C2, C8.128(C2×D4), C22.3(C4×D4), C4⋊1D4.17C4, C4⋊D4.18C4, C4⋊C8.232C22, (C2×C4).651C24, (C2×C8).404C23, C42.209(C2×C4), C4.4D4.15C4, (C4×D4).54C22, C4.197(C22×D4), C4⋊M4(2)⋊34C2, C23.35(C22×C4), C8⋊C4.155C22, C22⋊C8.140C22, C2.15(Q8○M4(2)), C22.178(C23×C4), (C2×C42).759C22, (C22×C4).918C23, (C22×C8).434C22, (C2×M4(2)).348C22, C22.26C24.25C2, C2.49(C2×C4×D4), (C2×C8○D4)⋊22C2, (C2×C8⋊C4)⋊34C2, C4⋊C4.115(C2×C4), C4.302(C2×C4○D4), (C2×C4).844(C2×D4), (C2×D4).137(C2×C4), C22⋊C4.16(C2×C4), (C2×Q8).155(C2×C4), (C22×C8)⋊C2⋊29C2, (C2×C4).684(C4○D4), (C22×C4).341(C2×C4), (C2×C4).262(C22×C4), (C2×C4○D4).286C22, SmallGroup(128,1664)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 242 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×2], C22×C8 [×4], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4 [×2], C2×C8⋊C4, (C22×C8)⋊C2 [×2], C4⋊M4(2), C8⋊9D4 [×8], C22.26C24, C2×C8○D4 [×2], C42.266C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○M4(2) [×2], C42.266C23
Generators and relations
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece=b2c, de=ed >
►On 64 points
(1 45 26 18)(2 42 27 23)(3 47 28 20)(4 44 29 17)(5 41 30 22)(6 46 31 19)(7 43 32 24)(8 48 25 21)(9 52 63 34)(10 49 64 39)(11 54 57 36)(12 51 58 33)(13 56 59 38)(14 53 60 35)(15 50 61 40)(16 55 62 37)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 57 13 61)(10 58 14 62)(11 59 15 63)(12 60 16 64)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(33 53 37 49)(34 54 38 50)(35 55 39 51)(36 56 40 52)
(1 39 5 35)(2 50 6 54)(3 33 7 37)(4 52 8 56)(9 48 13 44)(10 22 14 18)(11 42 15 46)(12 24 16 20)(17 63 21 59)(19 57 23 61)(25 38 29 34)(26 49 30 53)(27 40 31 36)(28 51 32 55)(41 60 45 64)(43 62 47 58)
(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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
G:=sub<Sym(64)| (1,45,26,18)(2,42,27,23)(3,47,28,20)(4,44,29,17)(5,41,30,22)(6,46,31,19)(7,43,32,24)(8,48,25,21)(9,52,63,34)(10,49,64,39)(11,54,57,36)(12,51,58,33)(13,56,59,38)(14,53,60,35)(15,50,61,40)(16,55,62,37), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,57,13,61)(10,58,14,62)(11,59,15,63)(12,60,16,64)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(33,53,37,49)(34,54,38,50)(35,55,39,51)(36,56,40,52), (1,39,5,35)(2,50,6,54)(3,33,7,37)(4,52,8,56)(9,48,13,44)(10,22,14,18)(11,42,15,46)(12,24,16,20)(17,63,21,59)(19,57,23,61)(25,38,29,34)(26,49,30,53)(27,40,31,36)(28,51,32,55)(41,60,45,64)(43,62,47,58), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59)>;
G:=Group( (1,45,26,18)(2,42,27,23)(3,47,28,20)(4,44,29,17)(5,41,30,22)(6,46,31,19)(7,43,32,24)(8,48,25,21)(9,52,63,34)(10,49,64,39)(11,54,57,36)(12,51,58,33)(13,56,59,38)(14,53,60,35)(15,50,61,40)(16,55,62,37), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,57,13,61)(10,58,14,62)(11,59,15,63)(12,60,16,64)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(33,53,37,49)(34,54,38,50)(35,55,39,51)(36,56,40,52), (1,39,5,35)(2,50,6,54)(3,33,7,37)(4,52,8,56)(9,48,13,44)(10,22,14,18)(11,42,15,46)(12,24,16,20)(17,63,21,59)(19,57,23,61)(25,38,29,34)(26,49,30,53)(27,40,31,36)(28,51,32,55)(41,60,45,64)(43,62,47,58), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59) );
G=PermutationGroup([(1,45,26,18),(2,42,27,23),(3,47,28,20),(4,44,29,17),(5,41,30,22),(6,46,31,19),(7,43,32,24),(8,48,25,21),(9,52,63,34),(10,49,64,39),(11,54,57,36),(12,51,58,33),(13,56,59,38),(14,53,60,35),(15,50,61,40),(16,55,62,37)], [(1,28,5,32),(2,29,6,25),(3,30,7,26),(4,31,8,27),(9,57,13,61),(10,58,14,62),(11,59,15,63),(12,60,16,64),(17,46,21,42),(18,47,22,43),(19,48,23,44),(20,41,24,45),(33,53,37,49),(34,54,38,50),(35,55,39,51),(36,56,40,52)], [(1,39,5,35),(2,50,6,54),(3,33,7,37),(4,52,8,56),(9,48,13,44),(10,22,14,18),(11,42,15,46),(12,24,16,20),(17,63,21,59),(19,57,23,61),(25,38,29,34),(26,49,30,53),(27,40,31,36),(28,51,32,55),(41,60,45,64),(43,62,47,58)], [(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)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 6 | 14 |
| 0 | 0 | 0 | 8 | 3 | 0 |
| 0 | 0 | 0 | 1 | 9 | 0 |
| 0 | 0 | 16 | 0 | 6 | 9 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 11 |
| 0 | 0 | 16 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 12 | 12 | 7 |
| 0 | 0 | 0 | 16 | 10 | 0 |
| 0 | 0 | 0 | 2 | 1 | 0 |
| 0 | 0 | 15 | 0 | 12 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,8,0,0,16,0,0,6,8,1,0,0,0,6,3,9,6,0,0,14,0,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,1,6,0,1,0,0,11,1,16,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,15,0,0,12,16,2,0,0,0,12,10,1,12,0,0,7,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,11,0,16,0,0,11,0,16,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 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 | C4 | C4 | C4 | C4 | D4 | C4○D4 | Q8○M4(2) |
| kernel | C42.266C23 | C2×C8⋊C4 | (C22×C8)⋊C2 | C4⋊M4(2) | C8⋊9D4 | C22.26C24 | C2×C8○D4 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C2×C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 8 | 4 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{266}C_2^3 % in TeX
G:=Group("C4^2.266C2^3"); // GroupNames label
G:=SmallGroup(128,1664);
// by ID
G=gap.SmallGroup(128,1664);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2019,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations