p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.310C23, C4.1302- (1+4), C4.1832+ (1+4), C8⋊9D4⋊51C2, C8⋊6D4⋊49C2, C8⋊4Q8⋊49C2, C4⋊D4.33C4, C22⋊Q8.33C4, C4⋊C8.242C22, (C2×C8).450C23, C42⋊2C2.7C4, (C4×C8).349C22, C42.240(C2×C4), (C2×C4).692C24, C4.4D4.26C4, (C4×D4).66C22, C42.C2.26C4, (C4×Q8).63C22, C4⋊M4(2)⋊40C2, C42.6C4⋊57C2, C23.49(C22×C4), C8⋊C4.109C22, C22⋊C8.151C22, C2.39(Q8○M4(2)), C22.214(C23×C4), (C22×C8).456C22, (C22×C4).952C23, (C2×C42).799C22, C22.D4.14C4, C42.6C22⋊35C2, C42.7C22⋊32C2, C42⋊C2.94C22, (C2×M4(2)).255C22, C23.36C23.19C2, C2.50(C23.33C23), C4⋊C4.125(C2×C4), (C2×D4).147(C2×C4), C22⋊C4.27(C2×C4), (C2×C4).90(C22×C4), (C2×Q8).130(C2×C4), (C22×C4).370(C2×C4), SmallGroup(128,1727)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 252 in 173 conjugacy classes, 124 normal (40 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×3], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×8], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C4⋊M4(2), C42.6C22 [×2], C42.6C4, C42.7C22 [×2], C8⋊9D4 [×4], C8⋊6D4 [×2], C8⋊4Q8 [×2], C23.36C23, C42.310C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4), 2- (1+4), C23.33C23, Q8○M4(2) [×2], C42.310C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=b2c, ece=a2b2c, ede=a2d >
►On 64 points
(1 39 51 14)(2 40 52 15)(3 33 53 16)(4 34 54 9)(5 35 55 10)(6 36 56 11)(7 37 49 12)(8 38 50 13)(17 45 29 57)(18 46 30 58)(19 47 31 59)(20 48 32 60)(21 41 25 61)(22 42 26 62)(23 43 27 63)(24 44 28 64)
(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)
(2 6)(4 8)(9 38)(10 35)(11 40)(12 37)(13 34)(14 39)(15 36)(16 33)(18 22)(20 24)(26 30)(28 32)(41 61)(42 58)(43 63)(44 60)(45 57)(46 62)(47 59)(48 64)(50 54)(52 56)
(1 47)(2 64)(3 41)(4 58)(5 43)(6 60)(7 45)(8 62)(9 26)(10 19)(11 28)(12 21)(13 30)(14 23)(15 32)(16 17)(18 38)(20 40)(22 34)(24 36)(25 37)(27 39)(29 33)(31 35)(42 50)(44 52)(46 54)(48 56)(49 57)(51 59)(53 61)(55 63)
G:=sub<Sym(64)| (1,39,51,14)(2,40,52,15)(3,33,53,16)(4,34,54,9)(5,35,55,10)(6,36,56,11)(7,37,49,12)(8,38,50,13)(17,45,29,57)(18,46,30,58)(19,47,31,59)(20,48,32,60)(21,41,25,61)(22,42,26,62)(23,43,27,63)(24,44,28,64), (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), (2,6)(4,8)(9,38)(10,35)(11,40)(12,37)(13,34)(14,39)(15,36)(16,33)(18,22)(20,24)(26,30)(28,32)(41,61)(42,58)(43,63)(44,60)(45,57)(46,62)(47,59)(48,64)(50,54)(52,56), (1,47)(2,64)(3,41)(4,58)(5,43)(6,60)(7,45)(8,62)(9,26)(10,19)(11,28)(12,21)(13,30)(14,23)(15,32)(16,17)(18,38)(20,40)(22,34)(24,36)(25,37)(27,39)(29,33)(31,35)(42,50)(44,52)(46,54)(48,56)(49,57)(51,59)(53,61)(55,63)>;
G:=Group( (1,39,51,14)(2,40,52,15)(3,33,53,16)(4,34,54,9)(5,35,55,10)(6,36,56,11)(7,37,49,12)(8,38,50,13)(17,45,29,57)(18,46,30,58)(19,47,31,59)(20,48,32,60)(21,41,25,61)(22,42,26,62)(23,43,27,63)(24,44,28,64), (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), (2,6)(4,8)(9,38)(10,35)(11,40)(12,37)(13,34)(14,39)(15,36)(16,33)(18,22)(20,24)(26,30)(28,32)(41,61)(42,58)(43,63)(44,60)(45,57)(46,62)(47,59)(48,64)(50,54)(52,56), (1,47)(2,64)(3,41)(4,58)(5,43)(6,60)(7,45)(8,62)(9,26)(10,19)(11,28)(12,21)(13,30)(14,23)(15,32)(16,17)(18,38)(20,40)(22,34)(24,36)(25,37)(27,39)(29,33)(31,35)(42,50)(44,52)(46,54)(48,56)(49,57)(51,59)(53,61)(55,63) );
G=PermutationGroup([(1,39,51,14),(2,40,52,15),(3,33,53,16),(4,34,54,9),(5,35,55,10),(6,36,56,11),(7,37,49,12),(8,38,50,13),(17,45,29,57),(18,46,30,58),(19,47,31,59),(20,48,32,60),(21,41,25,61),(22,42,26,62),(23,43,27,63),(24,44,28,64)], [(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)], [(2,6),(4,8),(9,38),(10,35),(11,40),(12,37),(13,34),(14,39),(15,36),(16,33),(18,22),(20,24),(26,30),(28,32),(41,61),(42,58),(43,63),(44,60),(45,57),(46,62),(47,59),(48,64),(50,54),(52,56)], [(1,47),(2,64),(3,41),(4,58),(5,43),(6,60),(7,45),(8,62),(9,26),(10,19),(11,28),(12,21),(13,30),(14,23),(15,32),(16,17),(18,38),(20,40),(22,34),(24,36),(25,37),(27,39),(29,33),(31,35),(42,50),(44,52),(46,54),(48,56),(49,57),(51,59),(53,61),(55,63)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 1 | 0 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 16 |
| 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 |
| 3 | 0 | 4 | 10 | 0 | 0 | 0 | 0 |
| 3 | 0 | 14 | 16 | 0 | 0 | 0 | 0 |
| 2 | 12 | 9 | 14 | 0 | 0 | 0 | 0 |
| 3 | 2 | 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 | 1 | 0 |
| 0 | 0 | 0 | 0 | 10 | 13 | 2 | 8 |
| 0 | 0 | 0 | 0 | 13 | 9 | 3 | 2 |
| 0 | 0 | 0 | 0 | 7 | 4 | 16 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 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 | 5 | 9 | 16 | 0 |
| 0 | 0 | 0 | 0 | 9 | 3 | 0 | 16 |
| 13 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 4 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 10 | 4 | 0 | 16 |
| 0 | 0 | 0 | 0 | 2 | 15 | 16 | 0 |
G:=sub<GL(8,GF(17))| [1,0,13,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,4,4,16,0,0,0,0,0,0,0,0,1,4,1,15,0,0,0,0,0,16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[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],[3,3,2,3,0,0,0,0,0,0,12,2,0,0,0,0,4,14,9,2,0,0,0,0,10,16,14,5,0,0,0,0,0,0,0,0,6,10,13,7,0,0,0,0,4,13,9,4,0,0,0,0,1,2,3,16,0,0,0,0,0,8,2,12],[1,0,13,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,5,9,0,0,0,0,0,1,9,3,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[13,16,16,0,0,0,0,0,0,0,0,1,0,0,0,0,15,4,4,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,16,10,2,0,0,0,0,15,13,4,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4O | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | 2+ (1+4) | 2- (1+4) | Q8○M4(2) |
| kernel | C42.310C23 | C4⋊M4(2) | C42.6C22 | C42.6C4 | C42.7C22 | C8⋊9D4 | C8⋊6D4 | C8⋊4Q8 | C23.36C23 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{310}C_2^3 % in TeX
G:=Group("C4^2.310C2^3"); // GroupNames label
G:=SmallGroup(128,1727);
// by ID
G=gap.SmallGroup(128,1727);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,1018,80,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,a*c=c*a,d*a*d=a^-1,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=b^2*c,e*c*e=a^2*b^2*c,e*d*e=a^2*d>;
// generators/relations