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, (C4×C8).109C22, (C2×C8).318C23, C4○3(C22⋊Q16), C4○3(Q8.D4), Q8.D4⋊54C2, C22⋊Q16⋊36C2, (C4×D4).75C22, (C2×D4).89C23, C23.671(C2×D4), (C22×C4).721D4, (C4×Q8).72C22, C22.30(C4○D8), C4○3(C22.D8), (C2×Q8).375C23, C22.D8.5C2, C4.Q8.154C22, C2.D8.173C22, C42.12C4⋊32C2, C4○3(C23.47D4), C23.47D4⋊37C2, C4⋊D4.163C22, 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
Subgroups: 332 in 195 conjugacy classes, 92 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×21], D4 [×4], Q8 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×4], C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×SD16 [×2], C2×Q16 [×2], C22×Q8, C42.12C4, C4×SD16 [×2], C4×Q16 [×2], Q8⋊D4, C22⋊Q16, Q8.D4 [×2], Q8.Q8 [×2], C22.D8, C23.47D4, C2×C4×Q8, C23.36C23, C42.226D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8.C22, C42.226D4
Generators and relations
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 >
►On 64 points
(1 63 5 59)(2 49 6 53)(3 57 7 61)(4 51 8 55)(9 21 13 17)(10 47 14 43)(11 23 15 19)(12 41 16 45)(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 11)(2 38 30 12)(3 39 31 13)(4 40 32 14)(5 33 25 15)(6 34 26 16)(7 35 27 9)(8 36 28 10)(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)(10 12)(11 15)(14 16)(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)(9,21,13,17)(10,47,14,43)(11,23,15,19)(12,41,16,45)(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), (1,37,29,11)(2,38,30,12)(3,39,31,13)(4,40,32,14)(5,33,25,15)(6,34,26,16)(7,35,27,9)(8,36,28,10)(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)(10,12)(11,15)(14,16)(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)(9,21,13,17)(10,47,14,43)(11,23,15,19)(12,41,16,45)(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), (1,37,29,11)(2,38,30,12)(3,39,31,13)(4,40,32,14)(5,33,25,15)(6,34,26,16)(7,35,27,9)(8,36,28,10)(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)(10,12)(11,15)(14,16)(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),(9,21,13,17),(10,47,14,43),(11,23,15,19),(12,41,16,45),(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)], [(1,37,29,11),(2,38,30,12),(3,39,31,13),(4,40,32,14),(5,33,25,15),(6,34,26,16),(7,35,27,9),(8,36,28,10),(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),(10,12),(11,15),(14,16),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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