p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.302C23, C4.1182- (1+4), C8⋊4Q8⋊39C2, (C4×Q8).31C4, C4⋊C8.237C22, (C2×C8).440C23, C42.228(C2×C4), (C2×C4).680C24, (C4×C8).339C22, (C2×C4).27M4(2), C4.16(C2×M4(2)), (C22×Q8).33C4, C8⋊C4.99C22, (C4×Q8).282C22, C22⋊C8.236C22, C2.33(Q8○M4(2)), C42.6C4.32C2, C23.231(C22×C4), C22.203(C23×C4), (C2×C42).787C22, C2.21(C22×M4(2)), C22.30(C2×M4(2)), C42.12C4.45C2, (C22×C4).1284C23, C2.23(C23.32C23), (C2×C4⋊C4).80C4, (C2×C4×Q8).46C2, C4⋊C4.231(C2×C4), (C2×Q8).211(C2×C4), (C22×C4).359(C2×C4), (C2×C4).278(C22×C4), SmallGroup(128,1715)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 228 in 178 conjugacy classes, 134 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×11], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×9], Q8 [×8], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×8], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×8], C22×Q8, C42.12C4 [×2], C42.6C4 [×4], C8⋊4Q8 [×8], C2×C4×Q8, C42.302C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C23×C4, 2- (1+4) [×2], C23.32C23, C22×M4(2), Q8○M4(2), C42.302C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=b2c, ede-1=a2d >
►On 64 points
(1 31 55 12)(2 9 56 28)(3 25 49 14)(4 11 50 30)(5 27 51 16)(6 13 52 32)(7 29 53 10)(8 15 54 26)(17 42 40 64)(18 61 33 47)(19 44 34 58)(20 63 35 41)(21 46 36 60)(22 57 37 43)(23 48 38 62)(24 59 39 45)
(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 63 55 41)(2 42 56 64)(3 57 49 43)(4 44 50 58)(5 59 51 45)(6 46 52 60)(7 61 53 47)(8 48 54 62)(9 17 28 40)(10 33 29 18)(11 19 30 34)(12 35 31 20)(13 21 32 36)(14 37 25 22)(15 23 26 38)(16 39 27 24)
(1 31 55 12)(2 28 56 9)(3 25 49 14)(4 30 50 11)(5 27 51 16)(6 32 52 13)(7 29 53 10)(8 26 54 15)(17 64 40 42)(18 61 33 47)(19 58 34 44)(20 63 35 41)(21 60 36 46)(22 57 37 43)(23 62 38 48)(24 59 39 45)
G:=sub<Sym(64)| (1,31,55,12)(2,9,56,28)(3,25,49,14)(4,11,50,30)(5,27,51,16)(6,13,52,32)(7,29,53,10)(8,15,54,26)(17,42,40,64)(18,61,33,47)(19,44,34,58)(20,63,35,41)(21,46,36,60)(22,57,37,43)(23,48,38,62)(24,59,39,45), (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), (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,17,28,40)(10,33,29,18)(11,19,30,34)(12,35,31,20)(13,21,32,36)(14,37,25,22)(15,23,26,38)(16,39,27,24), (1,31,55,12)(2,28,56,9)(3,25,49,14)(4,30,50,11)(5,27,51,16)(6,32,52,13)(7,29,53,10)(8,26,54,15)(17,64,40,42)(18,61,33,47)(19,58,34,44)(20,63,35,41)(21,60,36,46)(22,57,37,43)(23,62,38,48)(24,59,39,45)>;
G:=Group( (1,31,55,12)(2,9,56,28)(3,25,49,14)(4,11,50,30)(5,27,51,16)(6,13,52,32)(7,29,53,10)(8,15,54,26)(17,42,40,64)(18,61,33,47)(19,44,34,58)(20,63,35,41)(21,46,36,60)(22,57,37,43)(23,48,38,62)(24,59,39,45), (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), (1,63,55,41)(2,42,56,64)(3,57,49,43)(4,44,50,58)(5,59,51,45)(6,46,52,60)(7,61,53,47)(8,48,54,62)(9,17,28,40)(10,33,29,18)(11,19,30,34)(12,35,31,20)(13,21,32,36)(14,37,25,22)(15,23,26,38)(16,39,27,24), (1,31,55,12)(2,28,56,9)(3,25,49,14)(4,30,50,11)(5,27,51,16)(6,32,52,13)(7,29,53,10)(8,26,54,15)(17,64,40,42)(18,61,33,47)(19,58,34,44)(20,63,35,41)(21,60,36,46)(22,57,37,43)(23,62,38,48)(24,59,39,45) );
G=PermutationGroup([(1,31,55,12),(2,9,56,28),(3,25,49,14),(4,11,50,30),(5,27,51,16),(6,13,52,32),(7,29,53,10),(8,15,54,26),(17,42,40,64),(18,61,33,47),(19,44,34,58),(20,63,35,41),(21,46,36,60),(22,57,37,43),(23,48,38,62),(24,59,39,45)], [(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)], [(1,63,55,41),(2,42,56,64),(3,57,49,43),(4,44,50,58),(5,59,51,45),(6,46,52,60),(7,61,53,47),(8,48,54,62),(9,17,28,40),(10,33,29,18),(11,19,30,34),(12,35,31,20),(13,21,32,36),(14,37,25,22),(15,23,26,38),(16,39,27,24)], [(1,31,55,12),(2,28,56,9),(3,25,49,14),(4,30,50,11),(5,27,51,16),(6,32,52,13),(7,29,53,10),(8,26,54,15),(17,64,40,42),(18,61,33,47),(19,58,34,44),(20,63,35,41),(21,60,36,46),(22,57,37,43),(23,62,38,48),(24,59,39,45)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 3 | 14 | 16 | 1 |
| 0 | 0 | 3 | 0 | 15 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 8 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 3 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 12 | 0 | 13 | 0 |
| 0 | 0 | 12 | 0 | 9 | 4 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 14 | 1 | 16 |
| 0 | 0 | 14 | 0 | 2 | 16 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,3,3,0,0,15,16,14,0,0,0,0,0,16,15,0,0,0,0,1,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,8,0,0,0,0,9,14,0,0,0,0,0,0,14,0,12,0,0,0,0,0,0,16,0,0,2,0,3,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,12,12,0,0,0,13,0,0,0,0,0,0,13,9,0,0,0,0,0,4],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,14,0,0,15,16,14,0,0,0,0,0,1,2,0,0,0,0,16,16] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | 2- (1+4) | Q8○M4(2) |
| kernel | C42.302C23 | C42.12C4 | C42.6C4 | C8⋊4Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 4 | 8 | 1 | 6 | 8 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{302}C_2^3 % in TeX
G:=Group("C4^2.302C2^3"); // GroupNames label
G:=SmallGroup(128,1715);
// by ID
G=gap.SmallGroup(128,1715);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,219,100,675,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e^-1=b^2*c,e*d*e^-1=a^2*d>;
// generators/relations