p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.485C23, C4.712- (1+4), Q8.Q8⋊7C2, C4⋊C4.268D4, (C4×Q16)⋊25C2, (C8×D4).11C2, C8.5Q8⋊7C2, (C2×D4).241D4, C8.75(C4○D4), C2.55(Q8○D8), C8.18D4⋊23C2, C4⋊C8.320C22, C4⋊C4.241C23, (C4×C8).120C22, (C2×C8).196C23, (C2×C4).528C24, C22⋊C4.112D4, C23.114(C2×D4), C2.81(D4⋊6D4), (C4×D4).341C22, C22.10(C4○D8), C23.20D4⋊8C2, C23.25D4⋊9C2, (C2×Q8).234C23, (C4×Q8).171C22, C2.D8.194C22, C4.Q8.108C22, C22⋊Q8.97C22, C23.47D4⋊34C2, C22⋊C8.207C22, (C22×C8).196C22, (C2×Q16).138C22, C22.788(C22×D4), C42.C2.45C22, (C22×C4).1160C23, Q8⋊C4.118C22, C42⋊C2.200C22, C22.46C24.2C2, (C2×C2.D8)⋊29C2, C2.66(C2×C4○D8), C22⋊C4○(C2.D8), C4.110(C2×C4○D4), (C2×C4).931(C2×D4), (C2×C4⋊C4).680C22, SmallGroup(128,2068)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 280 in 171 conjugacy classes, 88 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×4], C23 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×4], C2.D8 [×3], C2.D8 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4, C4×Q8 [×2], C22⋊Q8 [×4], C22.D4 [×2], C42.C2 [×2], C42.C2 [×2], C42⋊2C2 [×2], C22×C8 [×2], C2×Q16, C2×C2.D8, C23.25D4, C8×D4, C4×Q16, C8.18D4 [×2], Q8.Q8 [×2], C23.47D4 [×2], C23.20D4 [×2], C8.5Q8, C22.46C24 [×2], C42.485C23
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, Q8○D8, C42.485C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=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 18 26 24)(2 19 27 21)(3 20 28 22)(4 17 25 23)(5 62 12 16)(6 63 9 13)(7 64 10 14)(8 61 11 15)(29 39 33 43)(30 40 34 44)(31 37 35 41)(32 38 36 42)(45 57 49 53)(46 58 50 54)(47 59 51 55)(48 60 52 56)
(1 54 28 60)(2 57 25 55)(3 56 26 58)(4 59 27 53)(5 40 10 42)(6 43 11 37)(7 38 12 44)(8 41 9 39)(13 29 61 35)(14 36 62 30)(15 31 63 33)(16 34 64 32)(17 47 21 49)(18 50 22 48)(19 45 23 51)(20 52 24 46)
(1 44 28 38)(2 41 25 39)(3 42 26 40)(4 43 27 37)(5 46 10 52)(6 47 11 49)(7 48 12 50)(8 45 9 51)(13 59 61 53)(14 60 62 54)(15 57 63 55)(16 58 64 56)(17 33 21 31)(18 34 22 32)(19 35 23 29)(20 36 24 30)
(1 24 26 18)(2 17 27 23)(3 22 28 20)(4 19 25 21)(5 64 12 14)(6 13 9 63)(7 62 10 16)(8 15 11 61)(29 41 33 37)(30 40 34 44)(31 43 35 39)(32 38 36 42)(45 57 49 53)(46 56 50 60)(47 59 51 55)(48 54 52 58)
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), (1,18,26,24)(2,19,27,21)(3,20,28,22)(4,17,25,23)(5,62,12,16)(6,63,9,13)(7,64,10,14)(8,61,11,15)(29,39,33,43)(30,40,34,44)(31,37,35,41)(32,38,36,42)(45,57,49,53)(46,58,50,54)(47,59,51,55)(48,60,52,56), (1,54,28,60)(2,57,25,55)(3,56,26,58)(4,59,27,53)(5,40,10,42)(6,43,11,37)(7,38,12,44)(8,41,9,39)(13,29,61,35)(14,36,62,30)(15,31,63,33)(16,34,64,32)(17,47,21,49)(18,50,22,48)(19,45,23,51)(20,52,24,46), (1,44,28,38)(2,41,25,39)(3,42,26,40)(4,43,27,37)(5,46,10,52)(6,47,11,49)(7,48,12,50)(8,45,9,51)(13,59,61,53)(14,60,62,54)(15,57,63,55)(16,58,64,56)(17,33,21,31)(18,34,22,32)(19,35,23,29)(20,36,24,30), (1,24,26,18)(2,17,27,23)(3,22,28,20)(4,19,25,21)(5,64,12,14)(6,13,9,63)(7,62,10,16)(8,15,11,61)(29,41,33,37)(30,40,34,44)(31,43,35,39)(32,38,36,42)(45,57,49,53)(46,56,50,60)(47,59,51,55)(48,54,52,58)>;
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), (1,18,26,24)(2,19,27,21)(3,20,28,22)(4,17,25,23)(5,62,12,16)(6,63,9,13)(7,64,10,14)(8,61,11,15)(29,39,33,43)(30,40,34,44)(31,37,35,41)(32,38,36,42)(45,57,49,53)(46,58,50,54)(47,59,51,55)(48,60,52,56), (1,54,28,60)(2,57,25,55)(3,56,26,58)(4,59,27,53)(5,40,10,42)(6,43,11,37)(7,38,12,44)(8,41,9,39)(13,29,61,35)(14,36,62,30)(15,31,63,33)(16,34,64,32)(17,47,21,49)(18,50,22,48)(19,45,23,51)(20,52,24,46), (1,44,28,38)(2,41,25,39)(3,42,26,40)(4,43,27,37)(5,46,10,52)(6,47,11,49)(7,48,12,50)(8,45,9,51)(13,59,61,53)(14,60,62,54)(15,57,63,55)(16,58,64,56)(17,33,21,31)(18,34,22,32)(19,35,23,29)(20,36,24,30), (1,24,26,18)(2,17,27,23)(3,22,28,20)(4,19,25,21)(5,64,12,14)(6,13,9,63)(7,62,10,16)(8,15,11,61)(29,41,33,37)(30,40,34,44)(31,43,35,39)(32,38,36,42)(45,57,49,53)(46,56,50,60)(47,59,51,55)(48,54,52,58) );
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)], [(1,18,26,24),(2,19,27,21),(3,20,28,22),(4,17,25,23),(5,62,12,16),(6,63,9,13),(7,64,10,14),(8,61,11,15),(29,39,33,43),(30,40,34,44),(31,37,35,41),(32,38,36,42),(45,57,49,53),(46,58,50,54),(47,59,51,55),(48,60,52,56)], [(1,54,28,60),(2,57,25,55),(3,56,26,58),(4,59,27,53),(5,40,10,42),(6,43,11,37),(7,38,12,44),(8,41,9,39),(13,29,61,35),(14,36,62,30),(15,31,63,33),(16,34,64,32),(17,47,21,49),(18,50,22,48),(19,45,23,51),(20,52,24,46)], [(1,44,28,38),(2,41,25,39),(3,42,26,40),(4,43,27,37),(5,46,10,52),(6,47,11,49),(7,48,12,50),(8,45,9,51),(13,59,61,53),(14,60,62,54),(15,57,63,55),(16,58,64,56),(17,33,21,31),(18,34,22,32),(19,35,23,29),(20,36,24,30)], [(1,24,26,18),(2,17,27,23),(3,22,28,20),(4,19,25,21),(5,64,12,14),(6,13,9,63),(7,62,10,16),(8,15,11,61),(29,41,33,37),(30,40,34,44),(31,43,35,39),(32,38,36,42),(45,57,49,53),(46,56,50,60),(47,59,51,55),(48,54,52,58)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,1,0,0,15,1],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,15,0,0,8,0,0,0,0,0,13,0,0,0,9,4],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,16,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | ··· | 4L | 4M | ··· | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 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 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 | 2- (1+4) | Q8○D8 |
| kernel | C42.485C23 | C2×C2.D8 | C23.25D4 | C8×D4 | C4×Q16 | C8.18D4 | Q8.Q8 | C23.47D4 | C23.20D4 | C8.5Q8 | C22.46C24 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{485}C_2^3 % in TeX
G:=Group("C4^2.485C2^3"); // GroupNames label
G:=SmallGroup(128,2068);
// by ID
G=gap.SmallGroup(128,2068);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^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