p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.359C23, C4⋊C4.352D4, Q8⋊Q8⋊8C2, D4⋊2Q8⋊8C2, D8⋊C4⋊12C2, (C4×SD16)⋊31C2, D4⋊D4.3C2, C4⋊C8.58C22, C4⋊C4.78C23, (C2×C8).52C23, Q16⋊C4⋊11C2, D4.15(C4○D4), D4.2D4⋊22C2, D4.7D4⋊21C2, (C2×C4).323C24, (C4×C8).262C22, Q8.13(C4○D4), Q8.D4⋊22C2, C22⋊C4.153D4, (C2×D8).61C22, (C4×D4).85C22, C23.262(C2×D4), C4⋊Q8.108C22, (C2×Q8).83C23, (C4×Q8).81C22, C8⋊C4.15C22, C4.Q8.22C22, C2.31(D4○SD16), (C2×D4).413C23, C4⋊D4.31C22, C23.46D4⋊7C2, C23.47D4⋊7C2, C22⋊C8.36C22, (C2×Q16).62C22, C22⋊Q8.31C22, D4⋊C4.38C22, (C22×C4).296C23, C42.7C22⋊8C2, C4.4D4.30C22, C22.583(C22×D4), C22.36C24⋊3C2, Q8⋊C4.176C22, (C2×SD16).146C22, C42⋊C2.134C22, C23.33C23⋊14C2, C2.124(C22.19C24), C4.208(C2×C4○D4), (C2×C4).507(C2×D4), (C2×C4⋊C4).617C22, (C2×C4○D4).146C22, SmallGroup(128,1857)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 192 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×8], Q8 [×2], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×D4 [×2], C4×Q8 [×3], C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16 [×2], C2×Q16, C2×C4○D4, C42.7C22, C4×SD16 [×2], Q16⋊C4, D8⋊C4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, Q8⋊Q8, D4⋊2Q8, C23.46D4, C23.47D4, C23.33C23, C22.36C24, C42.359C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○SD16 [×2], C42.359C23
Generators and relations
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2b2c, ece=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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 9 15 63)(6 10 16 64)(7 11 13 61)(8 12 14 62)(29 39 34 41)(30 40 35 42)(31 37 36 43)(32 38 33 44)(45 54 50 60)(46 55 51 57)(47 56 52 58)(48 53 49 59)
(1 49 20 48)(2 50 17 45)(3 51 18 46)(4 52 19 47)(5 31 15 36)(6 32 16 33)(7 29 13 34)(8 30 14 35)(9 43 63 37)(10 44 64 38)(11 41 61 39)(12 42 62 40)(21 57 26 55)(22 58 27 56)(23 59 28 53)(24 60 25 54)
(1 50 20 45)(2 46 17 51)(3 52 18 47)(4 48 19 49)(5 40 15 42)(6 43 16 37)(7 38 13 44)(8 41 14 39)(9 35 63 30)(10 31 64 36)(11 33 61 32)(12 29 62 34)(21 56 26 58)(22 59 27 53)(23 54 28 60)(24 57 25 55)
(1 32)(2 29)(3 30)(4 31)(5 58)(6 59)(7 60)(8 57)(9 52)(10 49)(11 50)(12 51)(13 54)(14 55)(15 56)(16 53)(17 34)(18 35)(19 36)(20 33)(21 40)(22 37)(23 38)(24 39)(25 41)(26 42)(27 43)(28 44)(45 61)(46 62)(47 63)(48 64)
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (1,49,20,48)(2,50,17,45)(3,51,18,46)(4,52,19,47)(5,31,15,36)(6,32,16,33)(7,29,13,34)(8,30,14,35)(9,43,63,37)(10,44,64,38)(11,41,61,39)(12,42,62,40)(21,57,26,55)(22,58,27,56)(23,59,28,53)(24,60,25,54), (1,50,20,45)(2,46,17,51)(3,52,18,47)(4,48,19,49)(5,40,15,42)(6,43,16,37)(7,38,13,44)(8,41,14,39)(9,35,63,30)(10,31,64,36)(11,33,61,32)(12,29,62,34)(21,56,26,58)(22,59,27,53)(23,54,28,60)(24,57,25,55), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64)>;
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,9,15,63)(6,10,16,64)(7,11,13,61)(8,12,14,62)(29,39,34,41)(30,40,35,42)(31,37,36,43)(32,38,33,44)(45,54,50,60)(46,55,51,57)(47,56,52,58)(48,53,49,59), (1,49,20,48)(2,50,17,45)(3,51,18,46)(4,52,19,47)(5,31,15,36)(6,32,16,33)(7,29,13,34)(8,30,14,35)(9,43,63,37)(10,44,64,38)(11,41,61,39)(12,42,62,40)(21,57,26,55)(22,58,27,56)(23,59,28,53)(24,60,25,54), (1,50,20,45)(2,46,17,51)(3,52,18,47)(4,48,19,49)(5,40,15,42)(6,43,16,37)(7,38,13,44)(8,41,14,39)(9,35,63,30)(10,31,64,36)(11,33,61,32)(12,29,62,34)(21,56,26,58)(22,59,27,53)(23,54,28,60)(24,57,25,55), (1,32)(2,29)(3,30)(4,31)(5,58)(6,59)(7,60)(8,57)(9,52)(10,49)(11,50)(12,51)(13,54)(14,55)(15,56)(16,53)(17,34)(18,35)(19,36)(20,33)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64) );
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)], [(1,28,20,23),(2,25,17,24),(3,26,18,21),(4,27,19,22),(5,9,15,63),(6,10,16,64),(7,11,13,61),(8,12,14,62),(29,39,34,41),(30,40,35,42),(31,37,36,43),(32,38,33,44),(45,54,50,60),(46,55,51,57),(47,56,52,58),(48,53,49,59)], [(1,49,20,48),(2,50,17,45),(3,51,18,46),(4,52,19,47),(5,31,15,36),(6,32,16,33),(7,29,13,34),(8,30,14,35),(9,43,63,37),(10,44,64,38),(11,41,61,39),(12,42,62,40),(21,57,26,55),(22,58,27,56),(23,59,28,53),(24,60,25,54)], [(1,50,20,45),(2,46,17,51),(3,52,18,47),(4,48,19,49),(5,40,15,42),(6,43,16,37),(7,38,13,44),(8,41,14,39),(9,35,63,30),(10,31,64,36),(11,33,61,32),(12,29,62,34),(21,56,26,58),(22,59,27,53),(23,54,28,60),(24,57,25,55)], [(1,32),(2,29),(3,30),(4,31),(5,58),(6,59),(7,60),(8,57),(9,52),(10,49),(11,50),(12,51),(13,54),(14,55),(15,56),(16,53),(17,34),(18,35),(19,36),(20,33),(21,40),(22,37),(23,38),(24,39),(25,41),(26,42),(27,43),(28,44),(45,61),(46,62),(47,63),(48,64)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 12 | 10 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,10,5,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,12,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 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 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D4 | D4○SD16 |
| kernel | C42.359C23 | C42.7C22 | C4×SD16 | Q16⋊C4 | D8⋊C4 | D4⋊D4 | D4.7D4 | D4.2D4 | Q8.D4 | Q8⋊Q8 | D4⋊2Q8 | C23.46D4 | C23.47D4 | C23.33C23 | C22.36C24 | C22⋊C4 | C4⋊C4 | D4 | Q8 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{359}C_2^3 % in TeX
G:=Group("C4^2.359C2^3"); // GroupNames label
G:=SmallGroup(128,1857);
// by ID
G=gap.SmallGroup(128,1857);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,1018,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations