p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.390C23, C8⋊Q8⋊8C2, C4⋊C4.250D4, (C4×Q16)⋊30C2, C8.2D4⋊7C2, C8⋊8D4.3C2, C8.D4⋊7C2, C8⋊D4.1C2, (C4×SD16)⋊17C2, C8.29(C4○D4), C2.26(Q8○D8), C22⋊C4.90D4, C23.87(C2×D4), Q16⋊C4⋊15C2, C8.18D4⋊27C2, C4⋊C4.117C23, (C2×C4).376C24, (C2×C8).278C23, (C4×C8).183C22, 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×C4).1056C23, C22.35C24⋊4C2, 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
Generators and relations for C42.390C23
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 >
Subgroups: 308 in 175 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, 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, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.26C24, D4○SD16, Q8○D8, C42.390C23
►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 50 44)(18 58 51 45)(19 59 52 46)(20 60 53 47)(21 61 54 48)(22 62 55 41)(23 63 56 42)(24 64 49 43)
(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 54)(18 49)(19 52)(20 55)(21 50)(22 53)(23 56)(24 51)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(41 60)(42 63)(43 58)(44 61)(45 64)(46 59)(47 62)(48 57)
(1 46)(2 43)(3 48)(4 45)(5 42)(6 47)(7 44)(8 41)(9 53)(10 50)(11 55)(12 52)(13 49)(14 54)(15 51)(16 56)(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)(17,57,50,44)(18,58,51,45)(19,59,52,46)(20,60,53,47)(21,61,54,48)(22,62,55,41)(23,63,56,42)(24,64,49,43), (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)(17,54)(18,49)(19,52)(20,55)(21,50)(22,53)(23,56)(24,51)(25,29)(26,32)(28,30)(33,39)(35,37)(36,40)(41,60)(42,63)(43,58)(44,61)(45,64)(46,59)(47,62)(48,57), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(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)(17,57,50,44)(18,58,51,45)(19,59,52,46)(20,60,53,47)(21,61,54,48)(22,62,55,41)(23,63,56,42)(24,64,49,43), (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)(17,54)(18,49)(19,52)(20,55)(21,50)(22,53)(23,56)(24,51)(25,29)(26,32)(28,30)(33,39)(35,37)(36,40)(41,60)(42,63)(43,58)(44,61)(45,64)(46,59)(47,62)(48,57), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(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),(17,57,50,44),(18,58,51,45),(19,59,52,46),(20,60,53,47),(21,61,54,48),(22,62,55,41),(23,63,56,42),(24,64,49,43)], [(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),(17,54),(18,49),(19,52),(20,55),(21,50),(22,53),(23,56),(24,51),(25,29),(26,32),(28,30),(33,39),(35,37),(36,40),(41,60),(42,63),(43,58),(44,61),(45,64),(46,59),(47,62),(48,57)], [(1,46),(2,43),(3,48),(4,45),(5,42),(6,47),(7,44),(8,41),(9,53),(10,50),(11,55),(12,52),(13,49),(14,54),(15,51),(16,56),(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)]])
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 |
Matrix representation of C42.390C23 ►in 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] >;
C42.390C23 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