p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.159D4, C23.34SD16, C22⋊C8⋊9C4, C4⋊C4.298D4, C4.137(C4×D4), C22⋊1(C4.Q8), C4.2(C22⋊Q8), (C22×C4).49Q8, C23.72(C4⋊C4), C2.2(Q8⋊D4), C23.758(C2×D4), (C22×C4).283D4, C22.4Q16⋊39C2, C2.2(C22⋊SD16), C22.78C22≀C2, C22.53(C2×SD16), C22.68(C8⋊C22), (C22×C8).311C22, (C23×C4).249C22, C23.7Q8.13C2, C2.9(C23.8Q8), (C22×C4).1350C23, C2.2(C23.47D4), C2.2(C23.46D4), C22.57(C8.C22), C2.11(M4(2)⋊C4), C22.82(C22.D4), (C2×C8)⋊18(C2×C4), C2.9(C2×C4.Q8), (C2×C4.Q8)⋊15C2, (C2×C4).52(C4⋊C4), (C2×C4).980(C2×D4), (C2×C4).200(C2×Q8), (C22×C4⋊C4).16C2, (C2×C22⋊C8).36C2, (C2×C4⋊C4).52C22, C22.110(C2×C4⋊C4), (C2×C4).746(C4○D4), (C2×C4).549(C22×C4), (C22×C4).272(C2×C4), SmallGroup(128,585)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.159D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 356 in 180 conjugacy classes, 72 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×C4.Q8, C22×C4⋊C4, C24.159D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×SD16, C8⋊C22, C8.C22, C23.8Q8, C2×C4.Q8, M4(2)⋊C4, Q8⋊D4, C22⋊SD16, C23.46D4, C23.47D4, C24.159D4
►On 64 points
(1 19)(2 58)(3 21)(4 60)(5 23)(6 62)(7 17)(8 64)(9 63)(10 18)(11 57)(12 20)(13 59)(14 22)(15 61)(16 24)(25 51)(26 44)(27 53)(28 46)(29 55)(30 48)(31 49)(32 42)(33 47)(34 56)(35 41)(36 50)(37 43)(38 52)(39 45)(40 54)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 44 19 38)(2 47 20 33)(3 42 21 36)(4 45 22 39)(5 48 23 34)(6 43 24 37)(7 46 17 40)(8 41 18 35)(9 54 63 28)(10 49 64 31)(11 52 57 26)(12 55 58 29)(13 50 59 32)(14 53 60 27)(15 56 61 30)(16 51 62 25)
G:=sub<Sym(64)| (1,19)(2,58)(3,21)(4,60)(5,23)(6,62)(7,17)(8,64)(9,63)(10,18)(11,57)(12,20)(13,59)(14,22)(15,61)(16,24)(25,51)(26,44)(27,53)(28,46)(29,55)(30,48)(31,49)(32,42)(33,47)(34,56)(35,41)(36,50)(37,43)(38,52)(39,45)(40,54), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,44,19,38)(2,47,20,33)(3,42,21,36)(4,45,22,39)(5,48,23,34)(6,43,24,37)(7,46,17,40)(8,41,18,35)(9,54,63,28)(10,49,64,31)(11,52,57,26)(12,55,58,29)(13,50,59,32)(14,53,60,27)(15,56,61,30)(16,51,62,25)>;
G:=Group( (1,19)(2,58)(3,21)(4,60)(5,23)(6,62)(7,17)(8,64)(9,63)(10,18)(11,57)(12,20)(13,59)(14,22)(15,61)(16,24)(25,51)(26,44)(27,53)(28,46)(29,55)(30,48)(31,49)(32,42)(33,47)(34,56)(35,41)(36,50)(37,43)(38,52)(39,45)(40,54), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,44,19,38)(2,47,20,33)(3,42,21,36)(4,45,22,39)(5,48,23,34)(6,43,24,37)(7,46,17,40)(8,41,18,35)(9,54,63,28)(10,49,64,31)(11,52,57,26)(12,55,58,29)(13,50,59,32)(14,53,60,27)(15,56,61,30)(16,51,62,25) );
G=PermutationGroup([[(1,19),(2,58),(3,21),(4,60),(5,23),(6,62),(7,17),(8,64),(9,63),(10,18),(11,57),(12,20),(13,59),(14,22),(15,61),(16,24),(25,51),(26,44),(27,53),(28,46),(29,55),(30,48),(31,49),(32,42),(33,47),(34,56),(35,41),(36,50),(37,43),(38,52),(39,45),(40,54)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,44,19,38),(2,47,20,33),(3,42,21,36),(4,45,22,39),(5,48,23,34),(6,43,24,37),(7,46,17,40),(8,41,18,35),(9,54,63,28),(10,49,64,31),(11,52,57,26),(12,55,58,29),(13,50,59,32),(14,53,60,27),(15,56,61,30),(16,51,62,25)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D4 | C4○D4 | SD16 | C8⋊C22 | C8.C22 |
| kernel | C24.159D4 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C2×C4.Q8 | C22×C4⋊C4 | C22⋊C8 | C4⋊C4 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 8 | 1 | 1 |
Matrix representation of C24.159D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 9 | 0 | 0 |
| 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 5 | 7 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 13 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,4,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,13,0,0,0,9,16,0,0,0,0,0,0,5,0,0,0,7,7],[4,0,0,0,0,0,13,0,0,0,0,15,4,0,0,0,0,0,4,4,0,0,0,0,13] >;
C24.159D4 in GAP, Magma, Sage, TeX
C_2^4._{159}D_4 % in TeX
G:=Group("C2^4.159D4"); // GroupNames label
G:=SmallGroup(128,585);
// by ID
G=gap.SmallGroup(128,585);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,2019,1018,248]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations