p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C43⋊15C2, C42⋊41D4, C23.767C24, C24.128C23, C4⋊1(C4⋊1D4), C22.473(C22×D4), (C2×C42).1098C22, (C22×C4).1487C23, (C22×D4).317C22, (C2×C4⋊1D4)⋊13C2, (C2×C4).837(C2×D4), C2.18(C2×C4⋊1D4), SmallGroup(128,1599)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43⋊15C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1476 in 636 conjugacy classes, 180 normal (4 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C42, C22×C4, C2×D4, C24, C2×C42, C4⋊1D4, C22×D4, C43, C2×C4⋊1D4, C43⋊15C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, C2×C4⋊1D4, C43⋊15C2
►On 64 points
Generators in S64
(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 17 9 45)(2 18 10 46)(3 19 11 47)(4 20 12 48)(5 60 39 30)(6 57 40 31)(7 58 37 32)(8 59 38 29)(13 49 41 21)(14 50 42 22)(15 51 43 23)(16 52 44 24)(25 63 55 34)(26 64 56 35)(27 61 53 36)(28 62 54 33)
(1 57 53 41)(2 58 54 42)(3 59 55 43)(4 60 56 44)(5 64 52 48)(6 61 49 45)(7 62 50 46)(8 63 51 47)(9 31 27 13)(10 32 28 14)(11 29 25 15)(12 30 26 16)(17 40 36 21)(18 37 33 22)(19 38 34 23)(20 39 35 24)
(1 49)(2 52)(3 51)(4 50)(5 54)(6 53)(7 56)(8 55)(9 21)(10 24)(11 23)(12 22)(13 17)(14 20)(15 19)(16 18)(25 38)(26 37)(27 40)(28 39)(29 34)(30 33)(31 36)(32 35)(41 45)(42 48)(43 47)(44 46)(57 61)(58 64)(59 63)(60 62)
G:=sub<Sym(64)| (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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,57,53,41)(2,58,54,42)(3,59,55,43)(4,60,56,44)(5,64,52,48)(6,61,49,45)(7,62,50,46)(8,63,51,47)(9,31,27,13)(10,32,28,14)(11,29,25,15)(12,30,26,16)(17,40,36,21)(18,37,33,22)(19,38,34,23)(20,39,35,24), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18)(25,38)(26,37)(27,40)(28,39)(29,34)(30,33)(31,36)(32,35)(41,45)(42,48)(43,47)(44,46)(57,61)(58,64)(59,63)(60,62)>;
G:=Group( (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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,57,53,41)(2,58,54,42)(3,59,55,43)(4,60,56,44)(5,64,52,48)(6,61,49,45)(7,62,50,46)(8,63,51,47)(9,31,27,13)(10,32,28,14)(11,29,25,15)(12,30,26,16)(17,40,36,21)(18,37,33,22)(19,38,34,23)(20,39,35,24), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18)(25,38)(26,37)(27,40)(28,39)(29,34)(30,33)(31,36)(32,35)(41,45)(42,48)(43,47)(44,46)(57,61)(58,64)(59,63)(60,62) );
G=PermutationGroup([[(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)], [(1,17,9,45),(2,18,10,46),(3,19,11,47),(4,20,12,48),(5,60,39,30),(6,57,40,31),(7,58,37,32),(8,59,38,29),(13,49,41,21),(14,50,42,22),(15,51,43,23),(16,52,44,24),(25,63,55,34),(26,64,56,35),(27,61,53,36),(28,62,54,33)], [(1,57,53,41),(2,58,54,42),(3,59,55,43),(4,60,56,44),(5,64,52,48),(6,61,49,45),(7,62,50,46),(8,63,51,47),(9,31,27,13),(10,32,28,14),(11,29,25,15),(12,30,26,16),(17,40,36,21),(18,37,33,22),(19,38,34,23),(20,39,35,24)], [(1,49),(2,52),(3,51),(4,50),(5,54),(6,53),(7,56),(8,55),(9,21),(10,24),(11,23),(12,22),(13,17),(14,20),(15,19),(16,18),(25,38),(26,37),(27,40),(28,39),(29,34),(30,33),(31,36),(32,35),(41,45),(42,48),(43,47),(44,46),(57,61),(58,64),(59,63),(60,62)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 2 |
| type | + | + | + | + |
| image | C1 | C2 | C2 | D4 |
| kernel | C43⋊15C2 | C43 | C2×C4⋊1D4 | C42 |
| # reps | 1 | 1 | 14 | 28 |
Matrix representation of C43⋊15C2 ►in GL6(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 2 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -2 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -2 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;
C43⋊15C2 in GAP, Magma, Sage, TeX
C_4^3\rtimes_{15}C_2 % in TeX
G:=Group("C4^3:15C2"); // GroupNames label
G:=SmallGroup(128,1599);
// by ID
G=gap.SmallGroup(128,1599);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations