p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.68D4, C42.149C23, C4.16C4≀C2, C4⋊Q8.17C4, C42.90(C2×C4), C42⋊C2.4C4, (C22×C4).739D4, C8⋊C4.147C22, C42.6C4.20C2, C42.2C22⋊8C2, (C2×C42).193C22, C42.C2.96C22, C23.105(C22⋊C4), C22.2(C4.10D4), C2.31(C42⋊C22), C23.37C23.10C2, C2.36(C2×C4≀C2), C4⋊C4.27(C2×C4), (C2×C8⋊C4).20C2, (C2×C4).1177(C2×D4), (C22×C4).215(C2×C4), (C2×C4).143(C22×C4), C2.12(C2×C4.10D4), (C2×C4).321(C22⋊C4), C22.207(C2×C22⋊C4), SmallGroup(128,263)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.68D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 188 in 106 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C42.2C22, C2×C8⋊C4, C42.6C4, C23.37C23, C42.68D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C4≀C2, C2×C22⋊C4, C2×C4.10D4, C2×C4≀C2, C42⋊C22, C42.68D4
►On 64 points
(1 12 20 58)(2 63 21 9)(3 14 22 60)(4 57 23 11)(5 16 24 62)(6 59 17 13)(7 10 18 64)(8 61 19 15)(25 34 52 44)(26 41 53 39)(27 36 54 46)(28 43 55 33)(29 38 56 48)(30 45 49 35)(31 40 50 42)(32 47 51 37)
(1 10 24 60)(2 15 17 57)(3 12 18 62)(4 9 19 59)(5 14 20 64)(6 11 21 61)(7 16 22 58)(8 13 23 63)(25 40 56 46)(26 37 49 43)(27 34 50 48)(28 39 51 45)(29 36 52 42)(30 33 53 47)(31 38 54 44)(32 35 55 41)
(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 38 10 54 24 44 60 31)(2 26 15 37 17 49 57 43)(3 46 12 25 18 40 62 56)(4 51 9 45 19 28 59 39)(5 34 14 50 20 48 64 27)(6 30 11 33 21 53 61 47)(7 42 16 29 22 36 58 52)(8 55 13 41 23 32 63 35)
G:=sub<Sym(64)| (1,12,20,58)(2,63,21,9)(3,14,22,60)(4,57,23,11)(5,16,24,62)(6,59,17,13)(7,10,18,64)(8,61,19,15)(25,34,52,44)(26,41,53,39)(27,36,54,46)(28,43,55,33)(29,38,56,48)(30,45,49,35)(31,40,50,42)(32,47,51,37), (1,10,24,60)(2,15,17,57)(3,12,18,62)(4,9,19,59)(5,14,20,64)(6,11,21,61)(7,16,22,58)(8,13,23,63)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,10,54,24,44,60,31)(2,26,15,37,17,49,57,43)(3,46,12,25,18,40,62,56)(4,51,9,45,19,28,59,39)(5,34,14,50,20,48,64,27)(6,30,11,33,21,53,61,47)(7,42,16,29,22,36,58,52)(8,55,13,41,23,32,63,35)>;
G:=Group( (1,12,20,58)(2,63,21,9)(3,14,22,60)(4,57,23,11)(5,16,24,62)(6,59,17,13)(7,10,18,64)(8,61,19,15)(25,34,52,44)(26,41,53,39)(27,36,54,46)(28,43,55,33)(29,38,56,48)(30,45,49,35)(31,40,50,42)(32,47,51,37), (1,10,24,60)(2,15,17,57)(3,12,18,62)(4,9,19,59)(5,14,20,64)(6,11,21,61)(7,16,22,58)(8,13,23,63)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,10,54,24,44,60,31)(2,26,15,37,17,49,57,43)(3,46,12,25,18,40,62,56)(4,51,9,45,19,28,59,39)(5,34,14,50,20,48,64,27)(6,30,11,33,21,53,61,47)(7,42,16,29,22,36,58,52)(8,55,13,41,23,32,63,35) );
G=PermutationGroup([[(1,12,20,58),(2,63,21,9),(3,14,22,60),(4,57,23,11),(5,16,24,62),(6,59,17,13),(7,10,18,64),(8,61,19,15),(25,34,52,44),(26,41,53,39),(27,36,54,46),(28,43,55,33),(29,38,56,48),(30,45,49,35),(31,40,50,42),(32,47,51,37)], [(1,10,24,60),(2,15,17,57),(3,12,18,62),(4,9,19,59),(5,14,20,64),(6,11,21,61),(7,16,22,58),(8,13,23,63),(25,40,56,46),(26,37,49,43),(27,34,50,48),(28,39,51,45),(29,36,52,42),(30,33,53,47),(31,38,54,44),(32,35,55,41)], [(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)], [(1,38,10,54,24,44,60,31),(2,26,15,37,17,49,57,43),(3,46,12,25,18,40,62,56),(4,51,9,45,19,28,59,39),(5,34,14,50,20,48,64,27),(6,30,11,33,21,53,61,47),(7,42,16,29,22,36,58,52),(8,55,13,41,23,32,63,35)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4≀C2 | C4.10D4 | C42⋊C22 |
| kernel | C42.68D4 | C42.2C22 | C2×C8⋊C4 | C42.6C4 | C23.37C23 | C42⋊C2 | C4⋊Q8 | C42 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.68D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 15 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 15 | 13 |
| 1 | 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 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 13 | 12 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 6 | 14 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 5 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 13 | 12 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,15,0,0,0,0,0,13,0,0,0,0,0,0,4,15,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,12,14,0,0,5,13,0,0,0,0,3,12,0,0],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,14,11,0,0,0,0,5,3,0,0,0,0,0,0,5,13,0,0,0,0,3,12] >;
C42.68D4 in GAP, Magma, Sage, TeX
C_4^2._{68}D_4 % in TeX
G:=Group("C4^2.68D4"); // GroupNames label
G:=SmallGroup(128,263);
// by ID
G=gap.SmallGroup(128,263);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,352,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations