p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.25D4, C8⋊C4.2C4, (C22×C8).2C4, C42.32(C2×C4), (C2×C4).34C42, (C22×C4).19Q8, C23.39(C4⋊C4), C4.9(C4.D4), (C22×C4).120D4, C2.9(C4.9C42), C4.9(C4.10D4), C42.6C4.7C2, C2.4(C4.C42), C22.1(C8.C4), (C2×C42).129C22, C2.4(C22.C42), C22.40(C2.C42), (C2×C4⋊C8).4C2, (C2×C4).17(C4⋊C4), (C22×C4).464(C2×C4), (C2×C4).296(C22⋊C4), SmallGroup(128,22)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.25D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=a-1b2, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >
Subgroups: 120 in 68 conjugacy classes, 34 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C2×C4⋊C8, C42.6C4, C42.25D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4.D4, C4.10D4, C8.C4, C4.9C42, C4.C42, C22.C42, C42.25D4
►On 64 points
Generators in S64
(1 3 5 7)(2 18 6 22)(4 20 8 24)(9 59 13 63)(10 12 14 16)(11 61 15 57)(17 19 21 23)(25 34 29 38)(26 28 30 32)(27 36 31 40)(33 35 37 39)(41 43 45 47)(42 55 46 51)(44 49 48 53)(50 52 54 56)(58 60 62 64)
(1 62 19 10)(2 11 20 63)(3 64 21 12)(4 13 22 57)(5 58 23 14)(6 15 24 59)(7 60 17 16)(8 9 18 61)(25 48 36 55)(26 56 37 41)(27 42 38 49)(28 50 39 43)(29 44 40 51)(30 52 33 45)(31 46 34 53)(32 54 35 47)
(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 31 3 40 5 27 7 36)(2 47 18 41 6 43 22 45)(4 52 20 54 8 56 24 50)(9 26 59 28 13 30 63 32)(10 46 12 51 14 42 16 55)(11 35 61 37 15 39 57 33)(17 25 19 34 21 29 23 38)(44 58 49 60 48 62 53 64)
G:=sub<Sym(64)| (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,59,13,63)(10,12,14,16)(11,61,15,57)(17,19,21,23)(25,34,29,38)(26,28,30,32)(27,36,31,40)(33,35,37,39)(41,43,45,47)(42,55,46,51)(44,49,48,53)(50,52,54,56)(58,60,62,64), (1,62,19,10)(2,11,20,63)(3,64,21,12)(4,13,22,57)(5,58,23,14)(6,15,24,59)(7,60,17,16)(8,9,18,61)(25,48,36,55)(26,56,37,41)(27,42,38,49)(28,50,39,43)(29,44,40,51)(30,52,33,45)(31,46,34,53)(32,54,35,47), (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), (1,31,3,40,5,27,7,36)(2,47,18,41,6,43,22,45)(4,52,20,54,8,56,24,50)(9,26,59,28,13,30,63,32)(10,46,12,51,14,42,16,55)(11,35,61,37,15,39,57,33)(17,25,19,34,21,29,23,38)(44,58,49,60,48,62,53,64)>;
G:=Group( (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,59,13,63)(10,12,14,16)(11,61,15,57)(17,19,21,23)(25,34,29,38)(26,28,30,32)(27,36,31,40)(33,35,37,39)(41,43,45,47)(42,55,46,51)(44,49,48,53)(50,52,54,56)(58,60,62,64), (1,62,19,10)(2,11,20,63)(3,64,21,12)(4,13,22,57)(5,58,23,14)(6,15,24,59)(7,60,17,16)(8,9,18,61)(25,48,36,55)(26,56,37,41)(27,42,38,49)(28,50,39,43)(29,44,40,51)(30,52,33,45)(31,46,34,53)(32,54,35,47), (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), (1,31,3,40,5,27,7,36)(2,47,18,41,6,43,22,45)(4,52,20,54,8,56,24,50)(9,26,59,28,13,30,63,32)(10,46,12,51,14,42,16,55)(11,35,61,37,15,39,57,33)(17,25,19,34,21,29,23,38)(44,58,49,60,48,62,53,64) );
G=PermutationGroup([[(1,3,5,7),(2,18,6,22),(4,20,8,24),(9,59,13,63),(10,12,14,16),(11,61,15,57),(17,19,21,23),(25,34,29,38),(26,28,30,32),(27,36,31,40),(33,35,37,39),(41,43,45,47),(42,55,46,51),(44,49,48,53),(50,52,54,56),(58,60,62,64)], [(1,62,19,10),(2,11,20,63),(3,64,21,12),(4,13,22,57),(5,58,23,14),(6,15,24,59),(7,60,17,16),(8,9,18,61),(25,48,36,55),(26,56,37,41),(27,42,38,49),(28,50,39,43),(29,44,40,51),(30,52,33,45),(31,46,34,53),(32,54,35,47)], [(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)], [(1,31,3,40,5,27,7,36),(2,47,18,41,6,43,22,45),(4,52,20,54,8,56,24,50),(9,26,59,28,13,30,63,32),(10,46,12,51,14,42,16,55),(11,35,61,37,15,39,57,33),(17,25,19,34,21,29,23,38),(44,58,49,60,48,62,53,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 8A | ··· | 8H | 8I | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | C8.C4 | C4.D4 | C4.10D4 | C4.9C42 |
| kernel | C42.25D4 | C2×C4⋊C8 | C42.6C4 | C8⋊C4 | C22×C8 | C42 | C22×C4 | C22×C4 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 2 | 1 | 1 | 8 | 1 | 1 | 2 |
Matrix representation of C42.25D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 | 3 |
| 0 | 0 | 1 | 0 | 8 | 14 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 8 |
| 0 | 0 | 0 | 4 | 0 | 15 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 9 | 7 |
| 0 | 0 | 16 | 6 | 3 | 9 |
| 0 | 0 | 10 | 10 | 0 | 16 |
| 0 | 0 | 13 | 7 | 0 | 7 |
| 9 | 12 | 0 | 0 | 0 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 1 | 10 |
| 0 | 0 | 8 | 0 | 0 | 16 |
| 0 | 0 | 16 | 1 | 0 | 3 |
| 0 | 0 | 15 | 0 | 0 | 15 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,8,16,15,0,0,3,14,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,8,15,0,13],[14,2,0,0,0,0,2,3,0,0,0,0,0,0,4,16,10,13,0,0,10,6,10,7,0,0,9,3,0,0,0,0,7,9,16,7],[9,12,0,0,0,0,12,8,0,0,0,0,0,0,2,8,16,15,0,0,0,0,1,0,0,0,1,0,0,0,0,0,10,16,3,15] >;
C42.25D4 in GAP, Magma, Sage, TeX
C_4^2._{25}D_4 % in TeX
G:=Group("C4^2.25D4"); // GroupNames label
G:=SmallGroup(128,22);
// by ID
G=gap.SmallGroup(128,22);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;
// generators/relations