p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.109D4, C23.10M4(2), (C2×C8)⋊24D4, C24.63(C2×C4), C4.49(C4⋊1D4), C2.21(C8⋊9D4), (C22×D4).32C4, C22.180(C4×D4), C4.204(C4⋊D4), C4.89(C4.4D4), C22.56(C8○D4), (C22×C8).55C22, (C23×C4).25C22, (C2×C42).314C22, C23.317(C22×C4), C22.70(C2×M4(2)), C2.19(C24.4C4), (C22×C4).1638C23, C2.7(C24.3C22), (C2×C4⋊C8)⋊16C2, (C2×C4×D4).23C2, (C2×C4⋊C4).61C4, (C2×C8⋊C4)⋊26C2, (C2×C22⋊C8)⋊38C2, (C2×C4).1544(C2×D4), (C2×C22⋊C4).41C4, (C2×C4).944(C4○D4), (C22×C4).128(C2×C4), (C2×C4).135(C22⋊C4), C22.277(C2×C22⋊C4), C2.28((C22×C8)⋊C2), SmallGroup(128,687)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.109D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 364 in 192 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C23×C4, C22×D4, C2×C8⋊C4, C2×C22⋊C8, C2×C4⋊C8, C2×C4×D4, C42.109D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C8○D4, C24.3C22, C24.4C4, (C22×C8)⋊C2, C8⋊9D4, C42.109D4
►On 64 points
(1 43 11 19)(2 48 12 24)(3 45 13 21)(4 42 14 18)(5 47 15 23)(6 44 16 20)(7 41 9 17)(8 46 10 22)(25 40 63 54)(26 37 64 51)(27 34 57 56)(28 39 58 53)(29 36 59 50)(30 33 60 55)(31 38 61 52)(32 35 62 49)
(1 27 5 31)(2 28 6 32)(3 29 7 25)(4 30 8 26)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(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)
(1 54 5 50)(2 20 6 24)(3 56 7 52)(4 22 8 18)(9 38 13 34)(10 42 14 46)(11 40 15 36)(12 44 16 48)(17 27 21 31)(19 29 23 25)(26 55 30 51)(28 49 32 53)(33 60 37 64)(35 62 39 58)(41 57 45 61)(43 59 47 63)
G:=sub<Sym(64)| (1,43,11,19)(2,48,12,24)(3,45,13,21)(4,42,14,18)(5,47,15,23)(6,44,16,20)(7,41,9,17)(8,46,10,22)(25,40,63,54)(26,37,64,51)(27,34,57,56)(28,39,58,53)(29,36,59,50)(30,33,60,55)(31,38,61,52)(32,35,62,49), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (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), (1,54,5,50)(2,20,6,24)(3,56,7,52)(4,22,8,18)(9,38,13,34)(10,42,14,46)(11,40,15,36)(12,44,16,48)(17,27,21,31)(19,29,23,25)(26,55,30,51)(28,49,32,53)(33,60,37,64)(35,62,39,58)(41,57,45,61)(43,59,47,63)>;
G:=Group( (1,43,11,19)(2,48,12,24)(3,45,13,21)(4,42,14,18)(5,47,15,23)(6,44,16,20)(7,41,9,17)(8,46,10,22)(25,40,63,54)(26,37,64,51)(27,34,57,56)(28,39,58,53)(29,36,59,50)(30,33,60,55)(31,38,61,52)(32,35,62,49), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(33,46,37,42)(34,47,38,43)(35,48,39,44)(36,41,40,45), (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), (1,54,5,50)(2,20,6,24)(3,56,7,52)(4,22,8,18)(9,38,13,34)(10,42,14,46)(11,40,15,36)(12,44,16,48)(17,27,21,31)(19,29,23,25)(26,55,30,51)(28,49,32,53)(33,60,37,64)(35,62,39,58)(41,57,45,61)(43,59,47,63) );
G=PermutationGroup([[(1,43,11,19),(2,48,12,24),(3,45,13,21),(4,42,14,18),(5,47,15,23),(6,44,16,20),(7,41,9,17),(8,46,10,22),(25,40,63,54),(26,37,64,51),(27,34,57,56),(28,39,58,53),(29,36,59,50),(30,33,60,55),(31,38,61,52),(32,35,62,49)], [(1,27,5,31),(2,28,6,32),(3,29,7,25),(4,30,8,26),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,54,21,50),(18,55,22,51),(19,56,23,52),(20,49,24,53),(33,46,37,42),(34,47,38,43),(35,48,39,44),(36,41,40,45)], [(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)], [(1,54,5,50),(2,20,6,24),(3,56,7,52),(4,22,8,18),(9,38,13,34),(10,42,14,46),(11,40,15,36),(12,44,16,48),(17,27,21,31),(19,29,23,25),(26,55,30,51),(28,49,32,53),(33,60,37,64),(35,62,39,58),(41,57,45,61),(43,59,47,63)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4○D4 | M4(2) | C8○D4 |
| kernel | C42.109D4 | C2×C8⋊C4 | C2×C22⋊C8 | C2×C4⋊C8 | C2×C4×D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22×D4 | C42 | C2×C8 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C42.109D4 ►in GL6(𝔽17)
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 9 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,9,0,0,0,0,13,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,15,8],[16,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;
C42.109D4 in GAP, Magma, Sage, TeX
C_4^2._{109}D_4 % in TeX
G:=Group("C4^2.109D4"); // GroupNames label
G:=SmallGroup(128,687);
// by ID
G=gap.SmallGroup(128,687);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations