p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.160D4, C23.20Q16, C23.35SD16, C4.59(C4×D4), C4⋊C4.308D4, C22⋊Q8⋊10C4, C23.764(C2×D4), (C22×C4).286D4, C22.31(C2×Q16), C4.125(C4⋊D4), C22.4Q16⋊10C2, C22.89C22≀C2, C2.4(C22⋊SD16), C22⋊1(Q8⋊C4), (C22×C8).31C22, C2.4(C22⋊Q16), C22.56(C2×SD16), C22.73(C8⋊C22), (C23×C4).254C22, (C22×Q8).11C22, C23.119(C22⋊C4), (C22×C4).1359C23, C2.3(C23.47D4), C2.3(C23.48D4), C22.62(C8.C22), C2.11(C23.23D4), C2.24(C23.36D4), C22.84(C22.D4), C4⋊C4⋊6(C2×C4), (C2×Q8)⋊5(C2×C4), (C2×Q8⋊C4)⋊3C2, (C2×C22⋊Q8).6C2, (C2×C4).1328(C2×D4), (C2×C22⋊C8).22C2, (C22×C4⋊C4).17C2, C2.20(C2×Q8⋊C4), (C2×C4).755(C4○D4), (C2×C4⋊C4).761C22, (C22×C4).276(C2×C4), (C2×C4).377(C22×C4), (C2×C4).130(C22⋊C4), C22.263(C2×C22⋊C4), SmallGroup(128,604)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.160D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >
Subgroups: 396 in 202 conjugacy classes, 72 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C22⋊C8, Q8⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C22×C8, C23×C4, C23×C4, C22×Q8, C22.4Q16, C2×C22⋊C8, C2×Q8⋊C4, C22×C4⋊C4, C2×C22⋊Q8, C24.160D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4○D4, Q8⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23.23D4, C2×Q8⋊C4, C23.36D4, C22⋊SD16, C22⋊Q16, C23.47D4, C23.48D4, C24.160D4
►On 64 points
(1 23)(2 30)(3 17)(4 32)(5 19)(6 26)(7 21)(8 28)(9 58)(10 43)(11 60)(12 45)(13 62)(14 47)(15 64)(16 41)(18 50)(20 52)(22 54)(24 56)(25 51)(27 53)(29 55)(31 49)(33 59)(34 44)(35 61)(36 46)(37 63)(38 48)(39 57)(40 42)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 46 5 42)(2 16 6 12)(3 44 7 48)(4 14 8 10)(9 23 13 19)(11 21 15 17)(18 47 22 43)(20 45 24 41)(25 40 29 36)(26 61 30 57)(27 38 31 34)(28 59 32 63)(33 50 37 54)(35 56 39 52)(49 60 53 64)(51 58 55 62)
G:=sub<Sym(64)| (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,58)(10,43)(11,60)(12,45)(13,62)(14,47)(15,64)(16,41)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(33,59)(34,44)(35,61)(36,46)(37,63)(38,48)(39,57)(40,42), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,46,5,42)(2,16,6,12)(3,44,7,48)(4,14,8,10)(9,23,13,19)(11,21,15,17)(18,47,22,43)(20,45,24,41)(25,40,29,36)(26,61,30,57)(27,38,31,34)(28,59,32,63)(33,50,37,54)(35,56,39,52)(49,60,53,64)(51,58,55,62)>;
G:=Group( (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,58)(10,43)(11,60)(12,45)(13,62)(14,47)(15,64)(16,41)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(33,59)(34,44)(35,61)(36,46)(37,63)(38,48)(39,57)(40,42), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,46,5,42)(2,16,6,12)(3,44,7,48)(4,14,8,10)(9,23,13,19)(11,21,15,17)(18,47,22,43)(20,45,24,41)(25,40,29,36)(26,61,30,57)(27,38,31,34)(28,59,32,63)(33,50,37,54)(35,56,39,52)(49,60,53,64)(51,58,55,62) );
G=PermutationGroup([[(1,23),(2,30),(3,17),(4,32),(5,19),(6,26),(7,21),(8,28),(9,58),(10,43),(11,60),(12,45),(13,62),(14,47),(15,64),(16,41),(18,50),(20,52),(22,54),(24,56),(25,51),(27,53),(29,55),(31,49),(33,59),(34,44),(35,61),(36,46),(37,63),(38,48),(39,57),(40,42)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,57),(40,58)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,46,5,42),(2,16,6,12),(3,44,7,48),(4,14,8,10),(9,23,13,19),(11,21,15,17),(18,47,22,43),(20,45,24,41),(25,40,29,36),(26,61,30,57),(27,38,31,34),(28,59,32,63),(33,50,37,54),(35,56,39,52),(49,60,53,64),(51,58,55,62)]])
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 | D4 | C4○D4 | SD16 | Q16 | C8⋊C22 | C8.C22 |
| kernel | C24.160D4 | C22.4Q16 | C2×C22⋊C8 | C2×Q8⋊C4 | C22×C4⋊C4 | C2×C22⋊Q8 | C22⋊Q8 | C4⋊C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C24.160D4 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 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 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 3 | 14 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[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],[13,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,14,3,0,0,0,14,14],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,4] >;
C24.160D4 in GAP, Magma, Sage, TeX
C_2^4._{160}D_4 % in TeX
G:=Group("C2^4.160D4"); // GroupNames label
G:=SmallGroup(128,604);
// by ID
G=gap.SmallGroup(128,604);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=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=b*e^3>;
// generators/relations