p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.486C23, C4.722- (1+4), (C8×D4)⋊24C2, C8⋊8D4⋊9C2, D4.Q8⋊7C2, Q8.Q8⋊8C2, C4⋊C4.269D4, C8.5Q8⋊8C2, (C4×SD16)⋊10C2, (C2×D4).242D4, C8.82(C4○D4), C4⋊C4.242C23, C4⋊C8.321C22, (C2×C4).529C24, (C4×C8).121C22, (C2×C8).361C23, C22⋊C4.113D4, C23.115(C2×D4), C2.82(D4⋊6D4), C2.D8.62C22, C2.87(D4○SD16), (C2×D4).250C23, (C4×D4).342C22, C22.13(C4○D8), C23.20D4⋊9C2, C23.48D4⋊9C2, C23.19D4⋊8C2, C4⋊D4.99C22, C22.D8⋊10C2, (C2×Q8).235C23, (C4×Q8).172C22, C4.Q8.169C22, C22⋊Q8.98C22, C23.25D4⋊10C2, C22⋊C8.208C22, (C22×C8).197C22, Q8⋊C4.17C22, C22.789(C22×D4), C42.C2.46C22, D4⋊C4.170C22, (C22×C4).1161C23, C22.46C24⋊8C2, (C2×SD16).165C22, C42⋊C2.201C22, C22.47C24.2C2, (C2×C4.Q8)⋊37C2, C2.67(C2×C4○D8), C22⋊C4○(C4.Q8), C4.111(C2×C4○D4), (C2×C4).932(C2×D4), (C2×C4⋊C4).681C22, SmallGroup(128,2069)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 320 in 177 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×7], Q8 [×2], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×4], SD16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8, C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8 [×5], C2.D8 [×4], C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×2], C42.C2, C42⋊2C2 [×2], C22×C8 [×2], C2×SD16, C2×C4.Q8, C23.25D4, C8×D4, C4×SD16, C8⋊8D4 [×2], D4.Q8, Q8.Q8, C22.D8, C23.19D4, C23.48D4, C23.20D4, C8.5Q8, C22.46C24, C22.47C24, C42.486C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C2×C4○D8, D4○SD16, C42.486C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >
►On 64 points
(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 20 39 23)(2 17 40 24)(3 18 37 21)(4 19 38 22)(5 10 61 29)(6 11 62 30)(7 12 63 31)(8 9 64 32)(13 53 26 58)(14 54 27 59)(15 55 28 60)(16 56 25 57)(33 43 49 46)(34 44 50 47)(35 41 51 48)(36 42 52 45)
(1 16 3 14)(2 28 4 26)(5 33 7 35)(6 52 8 50)(9 44 11 42)(10 46 12 48)(13 40 15 38)(17 55 19 53)(18 59 20 57)(21 54 23 56)(22 58 24 60)(25 37 27 39)(29 43 31 41)(30 45 32 47)(34 62 36 64)(49 63 51 61)
(1 34 37 52)(2 35 38 49)(3 36 39 50)(4 33 40 51)(5 58 63 55)(6 59 64 56)(7 60 61 53)(8 57 62 54)(9 25 30 14)(10 26 31 15)(11 27 32 16)(12 28 29 13)(17 48 22 43)(18 45 23 44)(19 46 24 41)(20 47 21 42)
(1 25 39 16)(2 15 40 28)(3 27 37 14)(4 13 38 26)(5 41 61 48)(6 47 62 44)(7 43 63 46)(8 45 64 42)(9 36 32 52)(10 51 29 35)(11 34 30 50)(12 49 31 33)(17 55 24 60)(18 59 21 54)(19 53 22 58)(20 57 23 56)
G:=sub<Sym(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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,10,61,29)(6,11,62,30)(7,12,63,31)(8,9,64,32)(13,53,26,58)(14,54,27,59)(15,55,28,60)(16,56,25,57)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,16,3,14)(2,28,4,26)(5,33,7,35)(6,52,8,50)(9,44,11,42)(10,46,12,48)(13,40,15,38)(17,55,19,53)(18,59,20,57)(21,54,23,56)(22,58,24,60)(25,37,27,39)(29,43,31,41)(30,45,32,47)(34,62,36,64)(49,63,51,61), (1,34,37,52)(2,35,38,49)(3,36,39,50)(4,33,40,51)(5,58,63,55)(6,59,64,56)(7,60,61,53)(8,57,62,54)(9,25,30,14)(10,26,31,15)(11,27,32,16)(12,28,29,13)(17,48,22,43)(18,45,23,44)(19,46,24,41)(20,47,21,42), (1,25,39,16)(2,15,40,28)(3,27,37,14)(4,13,38,26)(5,41,61,48)(6,47,62,44)(7,43,63,46)(8,45,64,42)(9,36,32,52)(10,51,29,35)(11,34,30,50)(12,49,31,33)(17,55,24,60)(18,59,21,54)(19,53,22,58)(20,57,23,56)>;
G:=Group( (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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,10,61,29)(6,11,62,30)(7,12,63,31)(8,9,64,32)(13,53,26,58)(14,54,27,59)(15,55,28,60)(16,56,25,57)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,16,3,14)(2,28,4,26)(5,33,7,35)(6,52,8,50)(9,44,11,42)(10,46,12,48)(13,40,15,38)(17,55,19,53)(18,59,20,57)(21,54,23,56)(22,58,24,60)(25,37,27,39)(29,43,31,41)(30,45,32,47)(34,62,36,64)(49,63,51,61), (1,34,37,52)(2,35,38,49)(3,36,39,50)(4,33,40,51)(5,58,63,55)(6,59,64,56)(7,60,61,53)(8,57,62,54)(9,25,30,14)(10,26,31,15)(11,27,32,16)(12,28,29,13)(17,48,22,43)(18,45,23,44)(19,46,24,41)(20,47,21,42), (1,25,39,16)(2,15,40,28)(3,27,37,14)(4,13,38,26)(5,41,61,48)(6,47,62,44)(7,43,63,46)(8,45,64,42)(9,36,32,52)(10,51,29,35)(11,34,30,50)(12,49,31,33)(17,55,24,60)(18,59,21,54)(19,53,22,58)(20,57,23,56) );
G=PermutationGroup([(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)], [(1,20,39,23),(2,17,40,24),(3,18,37,21),(4,19,38,22),(5,10,61,29),(6,11,62,30),(7,12,63,31),(8,9,64,32),(13,53,26,58),(14,54,27,59),(15,55,28,60),(16,56,25,57),(33,43,49,46),(34,44,50,47),(35,41,51,48),(36,42,52,45)], [(1,16,3,14),(2,28,4,26),(5,33,7,35),(6,52,8,50),(9,44,11,42),(10,46,12,48),(13,40,15,38),(17,55,19,53),(18,59,20,57),(21,54,23,56),(22,58,24,60),(25,37,27,39),(29,43,31,41),(30,45,32,47),(34,62,36,64),(49,63,51,61)], [(1,34,37,52),(2,35,38,49),(3,36,39,50),(4,33,40,51),(5,58,63,55),(6,59,64,56),(7,60,61,53),(8,57,62,54),(9,25,30,14),(10,26,31,15),(11,27,32,16),(12,28,29,13),(17,48,22,43),(18,45,23,44),(19,46,24,41),(20,47,21,42)], [(1,25,39,16),(2,15,40,28),(3,27,37,14),(4,13,38,26),(5,41,61,48),(6,47,62,44),(7,43,63,46),(8,45,64,42),(9,36,32,52),(10,51,29,35),(11,34,30,50),(12,49,31,33),(17,55,24,60),(18,59,21,54),(19,53,22,58),(20,57,23,56)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 3 | 14 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[0,13,0,0,13,0,0,0,0,0,4,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,3,3,0,0,3,14],[0,13,0,0,4,0,0,0,0,0,13,0,0,0,0,13] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- (1+4) | D4○SD16 |
| kernel | C42.486C23 | C2×C4.Q8 | C23.25D4 | C8×D4 | C4×SD16 | C8⋊8D4 | D4.Q8 | Q8.Q8 | C22.D8 | C23.19D4 | C23.48D4 | C23.20D4 | C8.5Q8 | C22.46C24 | C22.47C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{486}C_2^3 % in TeX
G:=Group("C4^2.486C2^3"); // GroupNames label
G:=SmallGroup(128,2069);
// by ID
G=gap.SmallGroup(128,2069);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations