p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.390C23, C8⋊Q8⋊8C2, C4⋊C4.250D4, (C4×Q16)⋊30C2, C8.2D4⋊7C2, C8⋊D4.1C2, C8.D4⋊7C2, C8⋊8D4.3C2, (C4×SD16)⋊17C2, C8.29(C4○D4), C22⋊C4.90D4, C2.26(Q8○D8), C23.87(C2×D4), Q16⋊C4⋊15C2, C8.18D4⋊27C2, C4⋊C4.117C23, (C4×C8).183C22, (C2×C8).278C23, (C2×C4).376C24, C4.SD16⋊43C2, (C4×D4).96C22, C4⋊Q8.118C22, SD16⋊C4⋊21C2, (C4×Q8).93C22, C8○2M4(2)⋊19C2, C4.Q8.28C22, C2.39(D4○SD16), (C2×D4).130C23, C4⋊D4.37C22, (C2×Q8).118C23, C8⋊C4.133C22, C2.D8.220C22, C22⋊Q8.37C22, (C22×C8).278C22, (C2×Q16).159C22, (C2×SD16).24C22, C4.4D4.36C22, C22.636(C22×D4), C42.C2.22C22, D4⋊C4.148C22, C22.35C24⋊4C2, (C22×C4).1056C23, Q8⋊C4.140C22, C42⋊C2.333C22, C42.78C22⋊29C2, (C2×M4(2)).286C22, C22.36C24.2C2, C2.73(C22.26C24), C4.61(C2×C4○D4), (C2×C4).148(C2×D4), SmallGroup(128,1910)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 308 in 175 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×3], Q8 [×7], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×4], Q16 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×3], C2×Q8, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×3], C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2 [×2], C42⋊2C2 [×3], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8○2M4(2), C4×SD16, C4×Q16, SD16⋊C4, Q16⋊C4, C8⋊8D4, C8.18D4, C8⋊D4, C8.D4, C4.SD16, C42.78C22, C8.2D4, C8⋊Q8, C22.35C24, C22.36C24, C42.390C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D4○SD16, Q8○D8, C42.390C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a2c, ece-1=b-1c, ede-1=b2d >
►On 64 points
(1 38 31 16)(2 39 32 9)(3 40 25 10)(4 33 26 11)(5 34 27 12)(6 35 28 13)(7 36 29 14)(8 37 30 15)(17 57 54 46)(18 58 55 47)(19 59 56 48)(20 60 49 41)(21 61 50 42)(22 62 51 43)(23 63 52 44)(24 64 53 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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 50)(18 53)(19 56)(20 51)(21 54)(22 49)(23 52)(24 55)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(41 62)(42 57)(43 60)(44 63)(45 58)(46 61)(47 64)(48 59)
(1 48)(2 45)(3 42)(4 47)(5 44)(6 41)(7 46)(8 43)(9 49)(10 54)(11 51)(12 56)(13 53)(14 50)(15 55)(16 52)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(32 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)
G:=sub<Sym(64)| (1,38,31,16)(2,39,32,9)(3,40,25,10)(4,33,26,11)(5,34,27,12)(6,35,28,13)(7,36,29,14)(8,37,30,15)(17,57,54,46)(18,58,55,47)(19,59,56,48)(20,60,49,41)(21,61,50,42)(22,62,51,43)(23,63,52,44)(24,64,53,45), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,29)(26,32)(28,30)(33,39)(35,37)(36,40)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59), (1,48)(2,45)(3,42)(4,47)(5,44)(6,41)(7,46)(8,43)(9,49)(10,54)(11,51)(12,56)(13,53)(14,50)(15,55)(16,52)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64), (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)>;
G:=Group( (1,38,31,16)(2,39,32,9)(3,40,25,10)(4,33,26,11)(5,34,27,12)(6,35,28,13)(7,36,29,14)(8,37,30,15)(17,57,54,46)(18,58,55,47)(19,59,56,48)(20,60,49,41)(21,61,50,42)(22,62,51,43)(23,63,52,44)(24,64,53,45), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,50)(18,53)(19,56)(20,51)(21,54)(22,49)(23,52)(24,55)(25,29)(26,32)(28,30)(33,39)(35,37)(36,40)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59), (1,48)(2,45)(3,42)(4,47)(5,44)(6,41)(7,46)(8,43)(9,49)(10,54)(11,51)(12,56)(13,53)(14,50)(15,55)(16,52)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64), (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) );
G=PermutationGroup([(1,38,31,16),(2,39,32,9),(3,40,25,10),(4,33,26,11),(5,34,27,12),(6,35,28,13),(7,36,29,14),(8,37,30,15),(17,57,54,46),(18,58,55,47),(19,59,56,48),(20,60,49,41),(21,61,50,42),(22,62,51,43),(23,63,52,44),(24,64,53,45)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,50),(18,53),(19,56),(20,51),(21,54),(22,49),(23,52),(24,55),(25,29),(26,32),(28,30),(33,39),(35,37),(36,40),(41,62),(42,57),(43,60),(44,63),(45,58),(46,61),(47,64),(48,59)], [(1,48),(2,45),(3,42),(4,47),(5,44),(6,41),(7,46),(8,43),(9,49),(10,54),(11,51),(12,56),(13,53),(14,50),(15,55),(16,52),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,61),(26,58),(27,63),(28,60),(29,57),(30,62),(31,59),(32,64)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 13 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 1 |
| 0 | 0 | 10 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 10 |
| 0 | 0 | 16 | 0 | 7 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,3,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[14,13,0,0,0,0,2,3,0,0,0,0,0,0,0,10,0,16,0,0,7,0,1,0,0,0,0,16,0,7,0,0,1,0,10,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | C42.390C23 | C8○2M4(2) | C4×SD16 | C4×Q16 | SD16⋊C4 | Q16⋊C4 | C8⋊8D4 | C8.18D4 | C8⋊D4 | C8.D4 | C4.SD16 | C42.78C22 | C8.2D4 | C8⋊Q8 | C22.35C24 | C22.36C24 | C22⋊C4 | C4⋊C4 | C8 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{390}C_2^3 % in TeX
G:=Group("C4^2.390C2^3"); // GroupNames label
G:=SmallGroup(128,1910);
// by ID
G=gap.SmallGroup(128,1910);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,1018,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=a^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d>;
// generators/relations