p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.26D4, C4⋊C8.1C4, (C4×C8).7C4, C4.30C4≀C2, C22⋊C8.2C4, (C2×C4).35C42, C42.33(C2×C4), (C22×C4).20Q8, C23.13(C4⋊C4), (C22×C4).179D4, C2.7(C42⋊6C4), (C4×M4(2)).10C2, C42.6C4.8C2, (C2×C42).130C22, C2.2(C4.10C42), C2.7(M4(2)⋊4C4), C22.41(C2.C42), (C2×C4).18(C4⋊C4), (C22×C4).92(C2×C4), (C2×C4).297(C22⋊C4), SmallGroup(128,23)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.26D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=ab-1, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=ac3 >
Subgroups: 120 in 69 conjugacy classes, 32 normal (24 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×4], C23, C42 [×4], C2×C8 [×6], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×3], C22⋊C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×M4(2), C4×M4(2), C42.6C4 [×2], C42.26D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×2], C4.10C42, C42⋊6C4, M4(2)⋊4C4, C42.26D4
►On 64 points
(1 42 31 9)(2 43 32 10)(3 44 25 11)(4 45 26 12)(5 46 27 13)(6 47 28 14)(7 48 29 15)(8 41 30 16)(17 40 52 61)(18 33 53 62)(19 34 54 63)(20 35 55 64)(21 36 56 57)(22 37 49 58)(23 38 50 59)(24 39 51 60)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 54 21 50)(18 51 22 55)(19 56 23 52)(20 53 24 49)(25 31 29 27)(26 28 30 32)(33 60 37 64)(34 57 38 61)(35 62 39 58)(36 59 40 63)(41 43 45 47)(42 48 46 44)
(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 62 44 55 27 37 15 24)(2 17 41 59 28 56 12 34)(3 39 46 18 29 64 9 49)(4 50 43 36 30 19 14 61)(5 58 48 51 31 33 11 20)(6 21 45 63 32 52 16 38)(7 35 42 22 25 60 13 53)(8 54 47 40 26 23 10 57)
G:=sub<Sym(64)| (1,42,31,9)(2,43,32,10)(3,44,25,11)(4,45,26,12)(5,46,27,13)(6,47,28,14)(7,48,29,15)(8,41,30,16)(17,40,52,61)(18,33,53,62)(19,34,54,63)(20,35,55,64)(21,36,56,57)(22,37,49,58)(23,38,50,59)(24,39,51,60), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,54,21,50)(18,51,22,55)(19,56,23,52)(20,53,24,49)(25,31,29,27)(26,28,30,32)(33,60,37,64)(34,57,38,61)(35,62,39,58)(36,59,40,63)(41,43,45,47)(42,48,46,44), (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), (1,62,44,55,27,37,15,24)(2,17,41,59,28,56,12,34)(3,39,46,18,29,64,9,49)(4,50,43,36,30,19,14,61)(5,58,48,51,31,33,11,20)(6,21,45,63,32,52,16,38)(7,35,42,22,25,60,13,53)(8,54,47,40,26,23,10,57)>;
G:=Group( (1,42,31,9)(2,43,32,10)(3,44,25,11)(4,45,26,12)(5,46,27,13)(6,47,28,14)(7,48,29,15)(8,41,30,16)(17,40,52,61)(18,33,53,62)(19,34,54,63)(20,35,55,64)(21,36,56,57)(22,37,49,58)(23,38,50,59)(24,39,51,60), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,54,21,50)(18,51,22,55)(19,56,23,52)(20,53,24,49)(25,31,29,27)(26,28,30,32)(33,60,37,64)(34,57,38,61)(35,62,39,58)(36,59,40,63)(41,43,45,47)(42,48,46,44), (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), (1,62,44,55,27,37,15,24)(2,17,41,59,28,56,12,34)(3,39,46,18,29,64,9,49)(4,50,43,36,30,19,14,61)(5,58,48,51,31,33,11,20)(6,21,45,63,32,52,16,38)(7,35,42,22,25,60,13,53)(8,54,47,40,26,23,10,57) );
G=PermutationGroup([(1,42,31,9),(2,43,32,10),(3,44,25,11),(4,45,26,12),(5,46,27,13),(6,47,28,14),(7,48,29,15),(8,41,30,16),(17,40,52,61),(18,33,53,62),(19,34,54,63),(20,35,55,64),(21,36,56,57),(22,37,49,58),(23,38,50,59),(24,39,51,60)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,54,21,50),(18,51,22,55),(19,56,23,52),(20,53,24,49),(25,31,29,27),(26,28,30,32),(33,60,37,64),(34,57,38,61),(35,62,39,58),(36,59,40,63),(41,43,45,47),(42,48,46,44)], [(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)], [(1,62,44,55,27,37,15,24),(2,17,41,59,28,56,12,34),(3,39,46,18,29,64,9,49),(4,50,43,36,30,19,14,61),(5,58,48,51,31,33,11,20),(6,21,45,63,32,52,16,38),(7,35,42,22,25,60,13,53),(8,54,47,40,26,23,10,57)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4J | 4K | 8A | ··· | 8H | 8I | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | Q8 | C4≀C2 | C4.10C42 | M4(2)⋊4C4 |
| kernel | C42.26D4 | C4×M4(2) | C42.6C4 | C4×C8 | C22⋊C8 | C4⋊C8 | C42 | C22×C4 | C22×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 1 | 1 | 8 | 2 | 2 |
Matrix representation of C42.26D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 13 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 9 |
| 0 | 0 | 0 | 0 | 13 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 5 |
| 0 | 0 | 0 | 0 | 11 | 3 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,13,0,0,0,0,9,1,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,5,0,0,0,14,11,0,0,0,0,5,3,0,0] >;
C42.26D4 in GAP, Magma, Sage, TeX
C_4^2._{26}D_4 % in TeX
G:=Group("C4^2.26D4"); // GroupNames label
G:=SmallGroup(128,23);
// by ID
G=gap.SmallGroup(128,23);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,248,3924,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a*b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*c^3>;
// generators/relations