p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.17C23, Q8.8(C2×D4), C4⋊C4.338D4, C2.8(Q8○D8), C4⋊2Q16⋊20C2, C4⋊C8.40C22, (C2×C8).19C23, (C2×Q8).126D4, (C22×Q16)⋊9C2, C4.75(C22×D4), C4.36(C4⋊D4), C4⋊C4.385C23, (C2×C4).248C24, Q8.D4⋊15C2, C22⋊C4.139D4, (C2×D4).54C23, C23.445(C2×D4), C4⋊Q8.100C22, (C4×Q8).64C22, (C2×Q8).41C23, C23.38D4⋊9C2, (C2×Q16).53C22, (C2×SD16).6C22, C22.83(C4⋊D4), (C22×C4).978C23, C42.6C22⋊6C2, (C22×C8).140C22, C23.24D4.3C2, C4.4D4.25C22, C22.508(C22×D4), D4⋊C4.156C22, C23.32C23⋊8C2, Q8⋊C4.146C22, (C2×M4(2)).55C22, (C22×Q8).275C22, C42⋊C2.103C22, C23.38C23.10C2, C4.158(C2×C4○D4), (C2×C4).468(C2×D4), C2.66(C2×C4⋊D4), (C2×C4).279(C4○D4), (C2×C8.C22).11C2, (C2×C4○D4).120C22, SmallGroup(128,1776)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 396 in 229 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×4], Q8 [×4], Q8 [×14], C23, C23, C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×4], Q16 [×12], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×4], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×4], C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×6], C2×Q16 [×4], C8.C22 [×4], C22×Q8 [×2], C2×C4○D4, C23.24D4, C23.38D4, C42.6C22, C4⋊2Q16 [×4], Q8.D4 [×4], C23.32C23, C23.38C23, C22×Q16, C2×C8.C22, C42.17C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, Q8○D8 [×2], C42.17C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=d2=b2, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, 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 4 39 38)(2 37 40 3)(5 9 61 32)(6 31 62 12)(7 11 63 30)(8 29 64 10)(13 25 26 16)(14 15 27 28)(17 18 24 21)(19 20 22 23)(33 47 49 44)(34 43 50 46)(35 45 51 42)(36 41 52 48)(53 56 58 57)(54 60 59 55)
(1 36 39 52)(2 33 40 49)(3 34 37 50)(4 35 38 51)(5 58 61 53)(6 59 62 54)(7 60 63 55)(8 57 64 56)(9 25 32 16)(10 26 29 13)(11 27 30 14)(12 28 31 15)(17 46 24 43)(18 47 21 44)(19 48 22 41)(20 45 23 42)
(1 27 37 16)(2 15 38 26)(3 25 39 14)(4 13 40 28)(5 43 63 48)(6 47 64 42)(7 41 61 46)(8 45 62 44)(9 36 30 50)(10 49 31 35)(11 34 32 52)(12 51 29 33)(17 55 22 58)(18 57 23 54)(19 53 24 60)(20 59 21 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), (1,4,39,38)(2,37,40,3)(5,9,61,32)(6,31,62,12)(7,11,63,30)(8,29,64,10)(13,25,26,16)(14,15,27,28)(17,18,24,21)(19,20,22,23)(33,47,49,44)(34,43,50,46)(35,45,51,42)(36,41,52,48)(53,56,58,57)(54,60,59,55), (1,36,39,52)(2,33,40,49)(3,34,37,50)(4,35,38,51)(5,58,61,53)(6,59,62,54)(7,60,63,55)(8,57,64,56)(9,25,32,16)(10,26,29,13)(11,27,30,14)(12,28,31,15)(17,46,24,43)(18,47,21,44)(19,48,22,41)(20,45,23,42), (1,27,37,16)(2,15,38,26)(3,25,39,14)(4,13,40,28)(5,43,63,48)(6,47,64,42)(7,41,61,46)(8,45,62,44)(9,36,30,50)(10,49,31,35)(11,34,32,52)(12,51,29,33)(17,55,22,58)(18,57,23,54)(19,53,24,60)(20,59,21,56)>;
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), (1,4,39,38)(2,37,40,3)(5,9,61,32)(6,31,62,12)(7,11,63,30)(8,29,64,10)(13,25,26,16)(14,15,27,28)(17,18,24,21)(19,20,22,23)(33,47,49,44)(34,43,50,46)(35,45,51,42)(36,41,52,48)(53,56,58,57)(54,60,59,55), (1,36,39,52)(2,33,40,49)(3,34,37,50)(4,35,38,51)(5,58,61,53)(6,59,62,54)(7,60,63,55)(8,57,64,56)(9,25,32,16)(10,26,29,13)(11,27,30,14)(12,28,31,15)(17,46,24,43)(18,47,21,44)(19,48,22,41)(20,45,23,42), (1,27,37,16)(2,15,38,26)(3,25,39,14)(4,13,40,28)(5,43,63,48)(6,47,64,42)(7,41,61,46)(8,45,62,44)(9,36,30,50)(10,49,31,35)(11,34,32,52)(12,51,29,33)(17,55,22,58)(18,57,23,54)(19,53,24,60)(20,59,21,56) );
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)], [(1,4,39,38),(2,37,40,3),(5,9,61,32),(6,31,62,12),(7,11,63,30),(8,29,64,10),(13,25,26,16),(14,15,27,28),(17,18,24,21),(19,20,22,23),(33,47,49,44),(34,43,50,46),(35,45,51,42),(36,41,52,48),(53,56,58,57),(54,60,59,55)], [(1,36,39,52),(2,33,40,49),(3,34,37,50),(4,35,38,51),(5,58,61,53),(6,59,62,54),(7,60,63,55),(8,57,64,56),(9,25,32,16),(10,26,29,13),(11,27,30,14),(12,28,31,15),(17,46,24,43),(18,47,21,44),(19,48,22,41),(20,45,23,42)], [(1,27,37,16),(2,15,38,26),(3,25,39,14),(4,13,40,28),(5,43,63,48),(6,47,64,42),(7,41,61,46),(8,45,62,44),(9,36,30,50),(10,49,31,35),(11,34,32,52),(12,51,29,33),(17,55,22,58),(18,57,23,54),(19,53,24,60),(20,59,21,56)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 14 | 15 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 4 |
| 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 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 13 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 | 10 | 7 |
| 0 | 0 | 5 | 0 | 5 | 7 |
| 14 | 15 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 4 | 0 | 4 |
G:=sub<GL(6,GF(17))| [14,5,0,0,0,0,15,3,0,0,0,0,0,0,0,4,13,13,0,0,0,13,0,4,0,0,4,4,0,0,0,0,0,9,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[14,4,0,0,0,0,15,3,0,0,0,0,0,0,0,4,13,13,0,0,0,13,0,0,0,0,13,4,0,13,0,0,0,9,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,5,5,0,0,5,12,12,0,0,0,0,0,10,5,0,0,0,0,7,7],[14,5,0,0,0,0,15,3,0,0,0,0,0,0,13,0,0,13,0,0,0,13,0,4,0,0,0,0,4,0,0,0,0,0,0,4] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | Q8○D8 |
| kernel | C42.17C23 | C23.24D4 | C23.38D4 | C42.6C22 | C4⋊2Q16 | Q8.D4 | C23.32C23 | C23.38C23 | C22×Q16 | C2×C8.C22 | C22⋊C4 | C4⋊C4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{17}C_2^3 % in TeX
G:=Group("C4^2.17C2^3"); // GroupNames label
G:=SmallGroup(128,1776);
// by ID
G=gap.SmallGroup(128,1776);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,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