p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.295D4, C4.32- 1+4, C42.429C23, C4.222+ 1+4, C8⋊8D4⋊5C2, D4.Q8⋊3C2, C4⋊D8⋊10C2, C8⋊7D4⋊19C2, C4.Q16⋊10C2, D4⋊2Q8⋊38C2, C4⋊SD16⋊39C2, D4.2D4⋊3C2, C4.116(C4○D8), C4⋊C4.186C23, C4⋊C8.292C22, (C2×C8).339C23, (C2×C4).445C24, (C2×D8).28C22, C23.404(C2×D4), (C22×C4).523D4, C4⋊Q8.324C22, C2.D8.44C22, C4.Q8.92C22, (C2×D4).188C23, D4⋊C4.6C22, (C4×D4).126C22, C22.3(C8⋊C22), (C4×Q8).122C22, (C2×Q8).175C23, Q8⋊C4.8C22, C4⋊D4.208C22, C4⋊1D4.176C22, (C22×C8).188C22, (C2×C42).902C22, (C2×SD16).88C22, C22.705(C22×D4), C22⋊Q8.212C22, (C22×C4).1578C23, C22.26C24⋊23C2, C4.4D4.163C22, C42.C2.140C22, C23.36C23⋊13C2, C2.64(C22.31C24), (C2×C4⋊C8)⋊30C2, C2.49(C2×C4○D8), (C2×C4).569(C2×D4), C2.66(C2×C8⋊C22), SmallGroup(128,1979)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.295D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=b-1, dcd=a2c3 >
Subgroups: 412 in 199 conjugacy classes, 88 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×C4○D4, C2×C4⋊C8, C4⋊D8, C4⋊SD16, D4.2D4, C8⋊8D4, C8⋊7D4, D4⋊2Q8, C4.Q16, D4.Q8, C23.36C23, C22.26C24, C42.295D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C8⋊C22, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C4○D8, C2×C8⋊C22, C42.295D4
►On 64 points
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)(41 54 45 50)(42 55 46 51)(43 56 47 52)(44 49 48 53)
(1 26 55 16)(2 9 56 27)(3 28 49 10)(4 11 50 29)(5 30 51 12)(6 13 52 31)(7 32 53 14)(8 15 54 25)(17 46 39 59)(18 60 40 47)(19 48 33 61)(20 62 34 41)(21 42 35 63)(22 64 36 43)(23 44 37 57)(24 58 38 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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 40)(24 39)(25 26)(27 32)(28 31)(29 30)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(64)| (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(41,54,45,50)(42,55,46,51)(43,56,47,52)(44,49,48,53), (1,26,55,16)(2,9,56,27)(3,28,49,10)(4,11,50,29)(5,30,51,12)(6,13,52,31)(7,32,53,14)(8,15,54,25)(17,46,39,59)(18,60,40,47)(19,48,33,61)(20,62,34,41)(21,42,35,63)(22,64,36,43)(23,44,37,57)(24,58,38,45), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,40)(24,39)(25,26)(27,32)(28,31)(29,30)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,30,21,26)(18,31,22,27)(19,32,23,28)(20,25,24,29)(41,54,45,50)(42,55,46,51)(43,56,47,52)(44,49,48,53), (1,26,55,16)(2,9,56,27)(3,28,49,10)(4,11,50,29)(5,30,51,12)(6,13,52,31)(7,32,53,14)(8,15,54,25)(17,46,39,59)(18,60,40,47)(19,48,33,61)(20,62,34,41)(21,42,35,63)(22,64,36,43)(23,44,37,57)(24,58,38,45), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,40)(24,39)(25,26)(27,32)(28,31)(29,30)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([[(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,40,13,36),(10,33,14,37),(11,34,15,38),(12,35,16,39),(17,30,21,26),(18,31,22,27),(19,32,23,28),(20,25,24,29),(41,54,45,50),(42,55,46,51),(43,56,47,52),(44,49,48,53)], [(1,26,55,16),(2,9,56,27),(3,28,49,10),(4,11,50,29),(5,30,51,12),(6,13,52,31),(7,32,53,14),(8,15,54,25),(17,46,39,59),(18,60,40,47),(19,48,33,61),(20,62,34,41),(21,42,35,63),(22,64,36,43),(23,44,37,57),(24,58,38,45)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,40),(24,39),(25,26),(27,32),(28,31),(29,30),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,64),(49,52),(50,51),(53,56),(54,55)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4O | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | 2+ 1+4 | 2- 1+4 | C8⋊C22 |
| kernel | C42.295D4 | C2×C4⋊C8 | C4⋊D8 | C4⋊SD16 | D4.2D4 | C8⋊8D4 | C8⋊7D4 | D4⋊2Q8 | C4.Q16 | D4.Q8 | C23.36C23 | C22.26C24 | C42 | C22×C4 | C4 | C4 | C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.295D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 6 |
| 0 | 0 | 0 | 0 | 6 | 13 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 11 |
| 0 | 0 | 0 | 0 | 11 | 4 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,4,6,0,0,0,0,6,13,0,0,0,0,0,0,4,6,0,0,0,0,6,13],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,4,6,0,0,0,0,6,13,0,0,0,0,0,0,13,11,0,0,0,0,11,4] >;
C42.295D4 in GAP, Magma, Sage, TeX
C_4^2._{295}D_4 % in TeX
G:=Group("C4^2.295D4"); // GroupNames label
G:=SmallGroup(128,1979);
// by ID
G=gap.SmallGroup(128,1979);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=a^2*c^3>;
// generators/relations