p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.505C23, C4.262- (1+4), (C8×Q8)⋊12C2, C4⋊C4.278D4, (C4×Q16)⋊16C2, Q8⋊3Q8⋊3C2, Q8.Q8⋊10C2, C4⋊2Q16⋊16C2, C4.76(C4○D8), (C4×C8).94C22, (C2×Q8).184D4, C2.62(Q8○D8), C4⋊C4.432C23, C4⋊C8.325C22, (C2×C8).207C23, (C2×C4).556C24, Q8.34(C4○D4), D4⋊Q8.10C2, C4.SD16⋊32C2, (C4×SD16).13C2, C4⋊Q8.185C22, Q8.D4.1C2, C2.64(Q8⋊5D4), (C4×D4).195C22, (C2×D4).268C23, (C2×Q8).254C23, (C4×Q8).187C22, C2.D8.202C22, C4.Q8.176C22, (C2×Q16).142C22, Q8⋊C4.19C22, C4.4D4.76C22, C22.816(C22×D4), C42.C2.61C22, D4⋊C4.152C22, (C2×SD16).171C22, C42.78C22.2C2, C22.50C24.8C2, C2.74(C2×C4○D8), C4.257(C2×C4○D4), (C2×C4).176(C2×D4), SmallGroup(128,2096)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 280 in 168 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×2], Q8 [×2], Q8 [×9], C23, C42, C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×2], C2×C8 [×2], SD16 [×2], Q16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8, C4×C8, C4×C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8 [×4], C4×Q8, C22⋊Q8, C4.4D4 [×2], C42.C2 [×2], C42.C2 [×2], C42⋊2C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C2×Q16 [×2], C4×SD16, C4×Q16 [×2], C8×Q8, C4⋊2Q16, Q8.D4 [×2], D4⋊Q8, Q8.Q8 [×2], C4.SD16, C42.78C22 [×2], C22.50C24, Q8⋊3Q8, C42.505C23
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), Q8⋊5D4, C2×C4○D8, Q8○D8, C42.505C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, 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 23 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 35 37 41)(30 36 38 42)(31 33 39 43)(32 34 40 44)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 21)(18 22)(19 23)(20 24)(29 35)(30 36)(31 33)(32 34)(37 41)(38 42)(39 43)(40 44)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 25 55)(2 60 26 54)(3 59 27 53)(4 58 28 56)(5 32 11 38)(6 31 12 37)(7 30 9 40)(8 29 10 39)(13 36 64 44)(14 35 61 43)(15 34 62 42)(16 33 63 41)(17 49 23 45)(18 52 24 48)(19 51 21 47)(20 50 22 46)
(1 29 27 37)(2 30 28 38)(3 31 25 39)(4 32 26 40)(5 60 9 56)(6 57 10 53)(7 58 11 54)(8 59 12 55)(13 52 62 46)(14 49 63 47)(15 50 64 48)(16 51 61 45)(17 33 21 43)(18 34 22 44)(19 35 23 41)(20 36 24 42)
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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,35,37,41)(30,36,38,42)(31,33,39,43)(32,34,40,44)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,21)(18,22)(19,23)(20,24)(29,35)(30,36)(31,33)(32,34)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,25,55)(2,60,26,54)(3,59,27,53)(4,58,28,56)(5,32,11,38)(6,31,12,37)(7,30,9,40)(8,29,10,39)(13,36,64,44)(14,35,61,43)(15,34,62,42)(16,33,63,41)(17,49,23,45)(18,52,24,48)(19,51,21,47)(20,50,22,46), (1,29,27,37)(2,30,28,38)(3,31,25,39)(4,32,26,40)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,33,21,43)(18,34,22,44)(19,35,23,41)(20,36,24,42)>;
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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,35,37,41)(30,36,38,42)(31,33,39,43)(32,34,40,44)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,21)(18,22)(19,23)(20,24)(29,35)(30,36)(31,33)(32,34)(37,41)(38,42)(39,43)(40,44)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,25,55)(2,60,26,54)(3,59,27,53)(4,58,28,56)(5,32,11,38)(6,31,12,37)(7,30,9,40)(8,29,10,39)(13,36,64,44)(14,35,61,43)(15,34,62,42)(16,33,63,41)(17,49,23,45)(18,52,24,48)(19,51,21,47)(20,50,22,46), (1,29,27,37)(2,30,28,38)(3,31,25,39)(4,32,26,40)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,33,21,43)(18,34,22,44)(19,35,23,41)(20,36,24,42) );
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)], [(1,23,27,19),(2,24,28,20),(3,21,25,17),(4,22,26,18),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,35,37,41),(30,36,38,42),(31,33,39,43),(32,34,40,44),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(5,13),(6,14),(7,15),(8,16),(9,62),(10,63),(11,64),(12,61),(17,21),(18,22),(19,23),(20,24),(29,35),(30,36),(31,33),(32,34),(37,41),(38,42),(39,43),(40,44),(45,49),(46,50),(47,51),(48,52),(53,55),(54,56),(57,59),(58,60)], [(1,57,25,55),(2,60,26,54),(3,59,27,53),(4,58,28,56),(5,32,11,38),(6,31,12,37),(7,30,9,40),(8,29,10,39),(13,36,64,44),(14,35,61,43),(15,34,62,42),(16,33,63,41),(17,49,23,45),(18,52,24,48),(19,51,21,47),(20,50,22,46)], [(1,29,27,37),(2,30,28,38),(3,31,25,39),(4,32,26,40),(5,60,9,56),(6,57,10,53),(7,58,11,54),(8,59,12,55),(13,52,62,46),(14,49,63,47),(15,50,64,48),(16,51,61,45),(17,33,21,43),(18,34,22,44),(19,35,23,41),(20,36,24,42)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 5 | 5 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0],[5,5,0,0,5,12,0,0,0,0,1,0,0,0,0,1] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4H | 4I | ··· | 4O | 4P | ··· | 4T | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2- (1+4) | Q8○D8 |
| kernel | C42.505C23 | C4×SD16 | C4×Q16 | C8×Q8 | C4⋊2Q16 | Q8.D4 | D4⋊Q8 | Q8.Q8 | C4.SD16 | C42.78C22 | C22.50C24 | Q8⋊3Q8 | C4⋊C4 | C2×Q8 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{505}C_2^3 % in TeX
G:=Group("C4^2.505C2^3"); // GroupNames label
G:=SmallGroup(128,2096);
// by ID
G=gap.SmallGroup(128,2096);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations