p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.305C23, C4.1222- (1+4), C4⋊Q8.37C4, C8⋊4Q8⋊43C2, C22⋊Q8.29C4, C4⋊C8.239C22, (C2×C8).444C23, (C4×C8).342C22, C42.232(C2×C4), (C2×C4).684C24, C42.C2.22C4, (C4×Q8).61C22, C8⋊C4.103C22, C22⋊C8.147C22, C2.35(Q8○M4(2)), C42.6C4.35C2, (C2×C42).791C22, C22.207(C23×C4), (C22×C4).948C23, C23.109(C22×C4), C42⋊C2.90C22, C42.7C22.5C2, C23.37C23.26C2, C2.27(C23.32C23), C4⋊C4.123(C2×C4), C22⋊C4.25(C2×C4), (C2×C4).86(C22×C4), (C2×Q8).128(C2×C4), (C22×C4).363(C2×C4), SmallGroup(128,1719)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 204 in 157 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×4], Q8 [×4], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C42.6C4 [×2], C42.7C22 [×4], C8⋊4Q8 [×8], C23.37C23, C42.305C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2- (1+4) [×2], C23.32C23, Q8○M4(2) [×2], C42.305C23
Generators and relations
G = < a,b,c,d,e | a4=b4=e2=1, c2=b, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=b2c, ece=a2c, de=ed >
►On 64 points
(1 63 55 16)(2 9 56 64)(3 57 49 10)(4 11 50 58)(5 59 51 12)(6 13 52 60)(7 61 53 14)(8 15 54 62)(17 35 31 45)(18 46 32 36)(19 37 25 47)(20 48 26 38)(21 39 27 41)(22 42 28 40)(23 33 29 43)(24 44 30 34)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 31 55 17)(2 28 56 22)(3 25 49 19)(4 30 50 24)(5 27 51 21)(6 32 52 18)(7 29 53 23)(8 26 54 20)(9 46 64 36)(10 43 57 33)(11 48 58 38)(12 45 59 35)(13 42 60 40)(14 47 61 37)(15 44 62 34)(16 41 63 39)
(2 56)(4 50)(6 52)(8 54)(9 60)(10 14)(11 62)(12 16)(13 64)(15 58)(18 32)(20 26)(22 28)(24 30)(33 37)(34 48)(35 39)(36 42)(38 44)(40 46)(41 45)(43 47)(57 61)(59 63)
G:=sub<Sym(64)| (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,35,31,45)(18,46,32,36)(19,37,25,47)(20,48,26,38)(21,39,27,41)(22,42,28,40)(23,33,29,43)(24,44,30,34), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,31,55,17)(2,28,56,22)(3,25,49,19)(4,30,50,24)(5,27,51,21)(6,32,52,18)(7,29,53,23)(8,26,54,20)(9,46,64,36)(10,43,57,33)(11,48,58,38)(12,45,59,35)(13,42,60,40)(14,47,61,37)(15,44,62,34)(16,41,63,39), (2,56)(4,50)(6,52)(8,54)(9,60)(10,14)(11,62)(12,16)(13,64)(15,58)(18,32)(20,26)(22,28)(24,30)(33,37)(34,48)(35,39)(36,42)(38,44)(40,46)(41,45)(43,47)(57,61)(59,63)>;
G:=Group( (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,35,31,45)(18,46,32,36)(19,37,25,47)(20,48,26,38)(21,39,27,41)(22,42,28,40)(23,33,29,43)(24,44,30,34), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,31,55,17)(2,28,56,22)(3,25,49,19)(4,30,50,24)(5,27,51,21)(6,32,52,18)(7,29,53,23)(8,26,54,20)(9,46,64,36)(10,43,57,33)(11,48,58,38)(12,45,59,35)(13,42,60,40)(14,47,61,37)(15,44,62,34)(16,41,63,39), (2,56)(4,50)(6,52)(8,54)(9,60)(10,14)(11,62)(12,16)(13,64)(15,58)(18,32)(20,26)(22,28)(24,30)(33,37)(34,48)(35,39)(36,42)(38,44)(40,46)(41,45)(43,47)(57,61)(59,63) );
G=PermutationGroup([(1,63,55,16),(2,9,56,64),(3,57,49,10),(4,11,50,58),(5,59,51,12),(6,13,52,60),(7,61,53,14),(8,15,54,62),(17,35,31,45),(18,46,32,36),(19,37,25,47),(20,48,26,38),(21,39,27,41),(22,42,28,40),(23,33,29,43),(24,44,30,34)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,31,55,17),(2,28,56,22),(3,25,49,19),(4,30,50,24),(5,27,51,21),(6,32,52,18),(7,29,53,23),(8,26,54,20),(9,46,64,36),(10,43,57,33),(11,48,58,38),(12,45,59,35),(13,42,60,40),(14,47,61,37),(15,44,62,34),(16,41,63,39)], [(2,56),(4,50),(6,52),(8,54),(9,60),(10,14),(11,62),(12,16),(13,64),(15,58),(18,32),(20,26),(22,28),(24,30),(33,37),(34,48),(35,39),(36,42),(38,44),(40,46),(41,45),(43,47),(57,61),(59,63)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 13 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 9 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 9 | 8 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 12 | 15 |
| 0 | 0 | 0 | 0 | 14 | 8 | 2 | 5 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 13 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,13,16,13,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[9,9,0,9,0,0,0,0,0,0,0,8,0,0,0,0,16,8,8,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4,14,0,14,0,0,0,0,4,13,2,8,0,0,0,0,0,0,12,2,0,0,0,0,0,0,15,5],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,16,4,0,0,0,0,0,15,13,16,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4Q | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | 2- (1+4) | Q8○M4(2) |
| kernel | C42.305C23 | C42.6C4 | C42.7C22 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C4 | C2 |
| # reps | 1 | 2 | 4 | 8 | 1 | 8 | 4 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{305}C_2^3 % in TeX
G:=Group("C4^2.305C2^3"); // GroupNames label
G:=SmallGroup(128,1719);
// by ID
G=gap.SmallGroup(128,1719);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,891,100,675,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations