p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.226D4, C42.702C23, (C4×Q16)⋊21C2, (C4×SD16)⋊4C2, C4○3(Q8.Q8), Q8.Q8⋊55C2, C4○3(Q8⋊D4), C4⋊C4.61C23, Q8.4(C4○D4), Q8⋊D4.5C2, C4⋊C8.310C22, (C2×C4).306C24, (C2×C8).318C23, (C4×C8).109C22, C4○3(Q8.D4), C4○3(C22⋊Q16), C22⋊Q16⋊36C2, Q8.D4⋊54C2, (C4×D4).75C22, (C2×D4).89C23, C23.671(C2×D4), (C22×C4).721D4, (C4×Q8).72C22, C4○3(C22.D8), C22.30(C4○D8), (C2×Q8).375C23, C22.D8.5C2, C2.D8.173C22, C42.12C4⋊32C2, C4.Q8.154C22, C4○3(C23.47D4), C4⋊D4.163C22, C23.47D4⋊37C2, C4.143(C8.C22), C22⋊C8.216C22, (C2×C42).833C22, (C2×Q16).122C22, C22.566(C22×D4), C22⋊Q8.168C22, D4⋊C4.162C22, (C22×C4).1022C23, Q8⋊C4.175C22, (C2×SD16).142C22, C4.4D4.129C22, C42.C2.106C22, (C22×Q8).477C22, C2.107(C22.19C24), C23.36C23.20C2, (C2×C4×Q8)⋊38C2, C2.27(C2×C4○D8), C4.191(C2×C4○D4), (C2×C4).495(C2×D4), C2.31(C2×C8.C22), (C2×C4⋊C4).935C22, SmallGroup(128,1840)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.226D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, dbd=a2b, dcd=c3 >
Subgroups: 332 in 195 conjugacy classes, 92 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×SD16, C2×Q16, C22×Q8, C42.12C4, C4×SD16, C4×Q16, Q8⋊D4, C22⋊Q16, Q8.D4, Q8.Q8, C22.D8, C23.47D4, C2×C4×Q8, C23.36C23, C42.226D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C8.C22, C22×D4, C2×C4○D4, C22.19C24, C2×C4○D8, C2×C8.C22, C42.226D4
►On 64 points
(1 63 5 59)(2 49 6 53)(3 57 7 61)(4 51 8 55)(9 19 13 23)(10 45 14 41)(11 21 15 17)(12 47 16 43)(18 36 22 40)(20 38 24 34)(25 52 29 56)(26 60 30 64)(27 54 31 50)(28 62 32 58)(33 44 37 48)(35 46 39 42)
(1 37 29 13)(2 38 30 14)(3 39 31 15)(4 40 32 16)(5 33 25 9)(6 34 26 10)(7 35 27 11)(8 36 28 12)(17 57 42 50)(18 58 43 51)(19 59 44 52)(20 60 45 53)(21 61 46 54)(22 62 47 55)(23 63 48 56)(24 64 41 49)
(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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 46)(18 41)(19 44)(20 47)(21 42)(22 45)(23 48)(24 43)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(49 62)(50 57)(51 60)(52 63)(53 58)(54 61)(55 64)(56 59)
G:=sub<Sym(64)| (1,63,5,59)(2,49,6,53)(3,57,7,61)(4,51,8,55)(9,19,13,23)(10,45,14,41)(11,21,15,17)(12,47,16,43)(18,36,22,40)(20,38,24,34)(25,52,29,56)(26,60,30,64)(27,54,31,50)(28,62,32,58)(33,44,37,48)(35,46,39,42), (1,37,29,13)(2,38,30,14)(3,39,31,15)(4,40,32,16)(5,33,25,9)(6,34,26,10)(7,35,27,11)(8,36,28,12)(17,57,42,50)(18,58,43,51)(19,59,44,52)(20,60,45,53)(21,61,46,54)(22,62,47,55)(23,63,48,56)(24,64,41,49), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59)>;
G:=Group( (1,63,5,59)(2,49,6,53)(3,57,7,61)(4,51,8,55)(9,19,13,23)(10,45,14,41)(11,21,15,17)(12,47,16,43)(18,36,22,40)(20,38,24,34)(25,52,29,56)(26,60,30,64)(27,54,31,50)(28,62,32,58)(33,44,37,48)(35,46,39,42), (1,37,29,13)(2,38,30,14)(3,39,31,15)(4,40,32,16)(5,33,25,9)(6,34,26,10)(7,35,27,11)(8,36,28,12)(17,57,42,50)(18,58,43,51)(19,59,44,52)(20,60,45,53)(21,61,46,54)(22,62,47,55)(23,63,48,56)(24,64,41,49), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,46)(18,41)(19,44)(20,47)(21,42)(22,45)(23,48)(24,43)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(49,62)(50,57)(51,60)(52,63)(53,58)(54,61)(55,64)(56,59) );
G=PermutationGroup([[(1,63,5,59),(2,49,6,53),(3,57,7,61),(4,51,8,55),(9,19,13,23),(10,45,14,41),(11,21,15,17),(12,47,16,43),(18,36,22,40),(20,38,24,34),(25,52,29,56),(26,60,30,64),(27,54,31,50),(28,62,32,58),(33,44,37,48),(35,46,39,42)], [(1,37,29,13),(2,38,30,14),(3,39,31,15),(4,40,32,16),(5,33,25,9),(6,34,26,10),(7,35,27,11),(8,36,28,12),(17,57,42,50),(18,58,43,51),(19,59,44,52),(20,60,45,53),(21,61,46,54),(22,62,47,55),(23,63,48,56),(24,64,41,49)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,46),(18,41),(19,44),(20,47),(21,42),(22,45),(23,48),(24,43),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38),(49,62),(50,57),(51,60),(52,63),(53,58),(54,61),(55,64),(56,59)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 4W | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C42.226D4 | C42.12C4 | C4×SD16 | C4×Q16 | Q8⋊D4 | C22⋊Q16 | Q8.D4 | Q8.Q8 | C22.D8 | C23.47D4 | C2×C4×Q8 | C23.36C23 | C42 | C22×C4 | Q8 | C22 | C4 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.226D4 ►in GL4(𝔽17) generated by
| 14 | 2 | 0 | 0 |
| 13 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 |
| 14 | 1 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 1 | 0 | 0 | 0 |
| 3 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [14,13,0,0,2,3,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,0,4,0,0,13,0],[16,14,0,0,0,1,0,0,0,0,5,5,0,0,12,5],[1,3,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C42.226D4 in GAP, Magma, Sage, TeX
C_4^2._{226}D_4 % in TeX
G:=Group("C4^2.226D4"); // GroupNames label
G:=SmallGroup(128,1840);
// by ID
G=gap.SmallGroup(128,1840);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations