p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.260C23, (C4×D4).23C4, (C4×Q8).22C4, C4.62(C8○D4), C4⋊C8.354C22, (C4×M4(2))⋊31C2, (C2×C4).641C24, C42.204(C2×C4), (C4×C8).325C22, (C2×C8).474C23, C8⋊C4.152C22, C42.12C4⋊46C2, C4.16(C42⋊C2), C22⋊C8.228C22, (C22×C4).912C23, C23.100(C22×C4), (C22×C8).508C22, C22.169(C23×C4), C4○2(C42.7C22), C4○2(C42.6C22), (C2×C42).1107C22, C42.6C22⋊36C2, C42.7C22⋊33C2, C22.4(C42⋊C2), C42⋊C2.290C22, (C2×M4(2)).343C22, C42○(C42.7C22), C42○(C42.6C22), (C2×C4×C8)⋊17C2, C2.11(C2×C8○D4), C4⋊C4.217(C2×C4), (C4×C4○D4).11C2, C4.292(C2×C4○D4), (C2×D4).227(C2×C4), C22⋊C4.68(C2×C4), (C2×Q8).205(C2×C4), C4○2((C22×C8)⋊C2), (C2×C4).679(C4○D4), (C2×C4).257(C22×C4), (C22×C4).335(C2×C4), C2.41(C2×C42⋊C2), (C22×C8)⋊C2.20C2, C42○((C22×C8)⋊C2), (C2×C4○D4).280C22, SmallGroup(128,1654)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 200 conjugacy classes, 136 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×8], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×8], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×6], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C2×C4×C8, C4×M4(2), (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.12C4 [×4], C42.7C22 [×4], C4×C4○D4, C42.260C23
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 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4 [×2], C42.260C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >
►On 64 points
(1 33 55 12)(2 13 56 34)(3 35 49 14)(4 15 50 36)(5 37 51 16)(6 9 52 38)(7 39 53 10)(8 11 54 40)(17 59 25 43)(18 44 26 60)(19 61 27 45)(20 46 28 62)(21 63 29 47)(22 48 30 64)(23 57 31 41)(24 42 32 58)
(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)
(1 21)(2 30)(3 23)(4 32)(5 17)(6 26)(7 19)(8 28)(9 60)(10 45)(11 62)(12 47)(13 64)(14 41)(15 58)(16 43)(18 52)(20 54)(22 56)(24 50)(25 51)(27 53)(29 55)(31 49)(33 63)(34 48)(35 57)(36 42)(37 59)(38 44)(39 61)(40 46)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
G:=sub<Sym(64)| (1,33,55,12)(2,13,56,34)(3,35,49,14)(4,15,50,36)(5,37,51,16)(6,9,52,38)(7,39,53,10)(8,11,54,40)(17,59,25,43)(18,44,26,60)(19,61,27,45)(20,46,28,62)(21,63,29,47)(22,48,30,64)(23,57,31,41)(24,42,32,58), (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), (1,21)(2,30)(3,23)(4,32)(5,17)(6,26)(7,19)(8,28)(9,60)(10,45)(11,62)(12,47)(13,64)(14,41)(15,58)(16,43)(18,52)(20,54)(22,56)(24,50)(25,51)(27,53)(29,55)(31,49)(33,63)(34,48)(35,57)(36,42)(37,59)(38,44)(39,61)(40,46), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)>;
G:=Group( (1,33,55,12)(2,13,56,34)(3,35,49,14)(4,15,50,36)(5,37,51,16)(6,9,52,38)(7,39,53,10)(8,11,54,40)(17,59,25,43)(18,44,26,60)(19,61,27,45)(20,46,28,62)(21,63,29,47)(22,48,30,64)(23,57,31,41)(24,42,32,58), (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), (1,21)(2,30)(3,23)(4,32)(5,17)(6,26)(7,19)(8,28)(9,60)(10,45)(11,62)(12,47)(13,64)(14,41)(15,58)(16,43)(18,52)(20,54)(22,56)(24,50)(25,51)(27,53)(29,55)(31,49)(33,63)(34,48)(35,57)(36,42)(37,59)(38,44)(39,61)(40,46), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60) );
G=PermutationGroup([(1,33,55,12),(2,13,56,34),(3,35,49,14),(4,15,50,36),(5,37,51,16),(6,9,52,38),(7,39,53,10),(8,11,54,40),(17,59,25,43),(18,44,26,60),(19,61,27,45),(20,46,28,62),(21,63,29,47),(22,48,30,64),(23,57,31,41),(24,42,32,58)], [(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)], [(1,21),(2,30),(3,23),(4,32),(5,17),(6,26),(7,19),(8,28),(9,60),(10,45),(11,62),(12,47),(13,64),(14,41),(15,58),(16,43),(18,52),(20,54),(22,56),(24,50),(25,51),(27,53),(29,55),(31,49),(33,63),(34,48),(35,57),(36,42),(37,59),(38,44),(39,61),(40,46)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[0,2,0,0,2,0,0,0,0,0,9,0,0,0,0,9],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 | C8○D4 |
| kernel | C42.260C23 | C2×C4×C8 | C4×M4(2) | (C22×C8)⋊C2 | C42.6C22 | C42.12C4 | C42.7C22 | C4×C4○D4 | C4×D4 | C4×Q8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 12 | 4 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{260}C_2^3 % in TeX
G:=Group("C4^2.260C2^3"); // GroupNames label
G:=SmallGroup(128,1654);
// by ID
G=gap.SmallGroup(128,1654);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,124]);
// 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,e*a*e=a*b^2,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