p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.298D4, C4.62- 1+4, C42.432C23, C4.252+ 1+4, Q8.Q8⋊4C2, D4.Q8⋊4C2, C8⋊8D4⋊30C2, C8⋊7D4.7C2, D4.2D4⋊4C2, C8.18D4⋊7C2, Q8.D4⋊4C2, C4⋊C4.189C23, C4⋊C8.316C22, (C2×C4).448C24, (C2×C8).173C23, (C2×D8).29C22, C22.5(C4○D8), C23.407(C2×D4), (C22×C4).174D4, C2.D8.47C22, C4.Q8.94C22, (C2×D4).190C23, (C4×D4).128C22, (C4×Q8).125C22, (C2×Q16).31C22, (C2×Q8).178C23, C4⋊D4.210C22, (C2×C42).905C22, (C22×C8).155C22, (C2×SD16).90C22, C22.708(C22×D4), C22⋊Q8.215C22, D4⋊C4.114C22, C2.71(D8⋊C22), (C22×C4).1581C23, Q8⋊C4.110C22, C4.4D4.165C22, C42.C2.142C22, C23.36C23⋊14C2, C2.67(C22.31C24), (C2×C4⋊C8)⋊32C2, C2.51(C2×C4○D8), (C2×C4).572(C2×D4), SmallGroup(128,1982)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.298D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=a-1b2, cbc-1=b-1, dbd=a2b-1, dcd=a2c3 >
Subgroups: 340 in 180 conjugacy classes, 86 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C2×C4⋊C8, D4.2D4, Q8.D4, C8⋊8D4, C8⋊7D4, C8.18D4, D4.Q8, Q8.Q8, C23.36C23, C42.298D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C4○D8, D8⋊C22, C42.298D4
►On 64 points
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)(41 52 45 56)(42 53 46 49)(43 54 47 50)(44 55 48 51)
(1 28 55 10)(2 11 56 29)(3 30 49 12)(4 13 50 31)(5 32 51 14)(6 15 52 25)(7 26 53 16)(8 9 54 27)(17 44 39 57)(18 58 40 45)(19 46 33 59)(20 60 34 47)(21 48 35 61)(22 62 36 41)(23 42 37 63)(24 64 38 43)
(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)| (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27)(41,52,45,56)(42,53,46,49)(43,54,47,50)(44,55,48,51), (1,28,55,10)(2,11,56,29)(3,30,49,12)(4,13,50,31)(5,32,51,14)(6,15,52,25)(7,26,53,16)(8,9,54,27)(17,44,39,57)(18,58,40,45)(19,46,33,59)(20,60,34,47)(21,48,35,61)(22,62,36,41)(23,42,37,63)(24,64,38,43), (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( (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27)(41,52,45,56)(42,53,46,49)(43,54,47,50)(44,55,48,51), (1,28,55,10)(2,11,56,29)(3,30,49,12)(4,13,50,31)(5,32,51,14)(6,15,52,25)(7,26,53,16)(8,9,54,27)(17,44,39,57)(18,58,40,45)(19,46,33,59)(20,60,34,47)(21,48,35,61)(22,62,36,41)(23,42,37,63)(24,64,38,43), (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([[(1,61,5,57),(2,62,6,58),(3,63,7,59),(4,64,8,60),(9,34,13,38),(10,35,14,39),(11,36,15,40),(12,37,16,33),(17,28,21,32),(18,29,22,25),(19,30,23,26),(20,31,24,27),(41,52,45,56),(42,53,46,49),(43,54,47,50),(44,55,48,51)], [(1,28,55,10),(2,11,56,29),(3,30,49,12),(4,13,50,31),(5,32,51,14),(6,15,52,25),(7,26,53,16),(8,9,54,27),(17,44,39,57),(18,58,40,45),(19,46,33,59),(20,60,34,47),(21,48,35,61),(22,62,36,41),(23,42,37,63),(24,64,38,43)], [(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 | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 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 | D4 | D4 | C4○D8 | 2+ 1+4 | 2- 1+4 | D8⋊C22 |
| kernel | C42.298D4 | C2×C4⋊C8 | D4.2D4 | Q8.D4 | C8⋊8D4 | C8⋊7D4 | C8.18D4 | D4.Q8 | Q8.Q8 | C23.36C23 | C42 | C22×C4 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.298D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 15 |
| 0 | 0 | 13 | 0 | 2 | 0 |
| 0 | 0 | 0 | 16 | 0 | 13 |
| 0 | 0 | 1 | 0 | 4 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 6 | 11 | 6 |
| 0 | 0 | 11 | 11 | 11 | 11 |
| 0 | 0 | 13 | 4 | 6 | 11 |
| 0 | 0 | 13 | 13 | 6 | 6 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 6 | 11 | 6 |
| 0 | 0 | 6 | 6 | 6 | 6 |
| 0 | 0 | 13 | 4 | 6 | 11 |
| 0 | 0 | 4 | 4 | 11 | 11 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,1,0,0,4,0,16,0,0,0,0,2,0,4,0,0,15,0,13,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,11,11,13,13,0,0,6,11,4,13,0,0,11,11,6,6,0,0,6,11,11,6],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,11,6,13,4,0,0,6,6,4,4,0,0,11,6,6,11,0,0,6,6,11,11] >;
C42.298D4 in GAP, Magma, Sage, TeX
C_4^2._{298}D_4 % in TeX
G:=Group("C4^2.298D4"); // GroupNames label
G:=SmallGroup(128,1982);
// by ID
G=gap.SmallGroup(128,1982);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,304,4037,1027,124]);
// 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^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations