p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.296D4, C4.42- (1+4), C42.430C23, C4.232+ (1+4), Q8.Q8⋊3C2, C8⋊8D4.1C2, Q8⋊Q8⋊40C2, D4⋊Q8⋊11C2, C4⋊2Q16⋊10C2, Q8.D4⋊3C2, D4.D4⋊40C2, C4.117(C4○D8), C8.18D4⋊19C2, C4⋊C4.187C23, C4⋊C8.340C22, (C2×C4).446C24, (C2×C8).171C23, (C22×C4).524D4, C23.405(C2×D4), C4⋊Q8.325C22, C2.D8.45C22, C4.Q8.93C22, (C4×D4).127C22, D4⋊C4.7C22, (C2×D4).189C23, (C2×Q16).29C22, (C2×Q8).176C23, (C4×Q8).123C22, C4⋊D4.209C22, (C2×C42).903C22, (C22×C8).189C22, (C2×SD16).89C22, C22.706(C22×D4), C22.3(C8.C22), C22⋊Q8.213C22, (C22×C4).1579C23, Q8⋊C4.109C22, C4.4D4.164C22, C42.C2.141C22, C23.37C23⋊25C2, C23.36C23.30C2, C2.65(C22.31C24), (C2×C4⋊C8)⋊31C2, C2.50(C2×C4○D8), (C2×C4).570(C2×D4), C2.66(C2×C8.C22), SmallGroup(128,1980)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 177 conjugacy classes, 88 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×C4⋊C8, D4.D4, C4⋊2Q16, Q8.D4 [×2], C8⋊8D4 [×2], C8.18D4 [×2], D4⋊Q8, Q8⋊Q8, Q8.Q8 [×2], C23.36C23, C23.37C23, C42.296D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C2×C4○D8, C2×C8.C22, C42.296D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=c3 >
►On 64 points
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(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 30 51 9)(2 10 52 31)(3 32 53 11)(4 12 54 25)(5 26 55 13)(6 14 56 27)(7 28 49 15)(8 16 50 29)(17 46 38 59)(18 60 39 47)(19 48 40 61)(20 62 33 41)(21 42 34 63)(22 64 35 43)(23 44 36 57)(24 58 37 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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 33 21 37)(18 36 22 40)(19 39 23 35)(20 34 24 38)(25 26 29 30)(27 32 31 28)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)
G:=sub<Sym(64)| (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(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), (1,30,51,9)(2,10,52,31)(3,32,53,11)(4,12,54,25)(5,26,55,13)(6,14,56,27)(7,28,49,15)(8,16,50,29)(17,46,38,59)(18,60,39,47)(19,48,40,61)(20,62,33,41)(21,42,34,63)(22,64,35,43)(23,44,36,57)(24,58,37,45), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55)>;
G:=Group( (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(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), (1,30,51,9)(2,10,52,31)(3,32,53,11)(4,12,54,25)(5,26,55,13)(6,14,56,27)(7,28,49,15)(8,16,50,29)(17,46,38,59)(18,60,39,47)(19,48,40,61)(20,62,33,41)(21,42,34,63)(22,64,35,43)(23,44,36,57)(24,58,37,45), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55) );
G=PermutationGroup([(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,34,13,38),(10,35,14,39),(11,36,15,40),(12,37,16,33),(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)], [(1,30,51,9),(2,10,52,31),(3,32,53,11),(4,12,54,25),(5,26,55,13),(6,14,56,27),(7,28,49,15),(8,16,50,29),(17,46,38,59),(18,60,39,47),(19,48,40,61),(20,62,33,41),(21,42,34,63),(22,64,35,43),(23,44,36,57),(24,58,37,45)], [(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)], [(1,4,5,8),(2,7,6,3),(9,12,13,16),(10,15,14,11),(17,33,21,37),(18,36,22,40),(19,39,23,35),(20,34,24,38),(25,26,29,30),(27,32,31,28),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60),(49,56,53,52),(50,51,54,55)])
Matrix representation ►G ⊆ 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 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 0 | 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 0 | 0 | 11 | 15 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[7,5,0,0,0,0,7,0,0,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,6,7,0,0,0,0,2,11,0,0],[7,5,0,0,0,0,7,10,0,0,0,0,0,0,0,0,11,10,0,0,0,0,15,6,0,0,6,7,0,0,0,0,2,11,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 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 | C42.296D4 | C2×C4⋊C8 | D4.D4 | C4⋊2Q16 | Q8.D4 | C8⋊8D4 | C8.18D4 | D4⋊Q8 | Q8⋊Q8 | Q8.Q8 | C23.36C23 | C23.37C23 | 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 |
In GAP, Magma, Sage, TeX
C_4^2._{296}D_4 % in TeX
G:=Group("C4^2.296D4"); // GroupNames label
G:=SmallGroup(128,1980);
// by ID
G=gap.SmallGroup(128,1980);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,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=c^3>;
// generators/relations