p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.73D4, C4.60(C4×D4), C4⋊C4.309D4, C22⋊Q8⋊11C4, (C22×C4).287D4, C23.765(C2×D4), C4.126(C4⋊D4), C22.4Q16⋊11C2, C22.90C22≀C2, C2.5(D4.7D4), C22.53(C4○D8), (C22×C8).32C22, C23.79(C22⋊C4), (C23×C4).255C22, (C22×Q8).12C22, (C22×C4).1360C23, C2.3(C23.20D4), C22.63(C8.C22), C2.20(C23.38D4), C2.24(C23.24D4), C2.12(C23.23D4), C22.85(C22.D4), C4⋊C4.69(C2×C4), (C2×Q8⋊C4)⋊4C2, (C2×C4).993(C2×D4), (C2×Q8).62(C2×C4), (C2×C22⋊Q8).7C2, (C2×C22⋊C8).23C2, (C2×C4).756(C4○D4), (C2×C4⋊C4).762C22, (C22×C4).277(C2×C4), (C2×C4).378(C22×C4), (C2×C4).191(C22⋊C4), (C2×C42⋊C2).20C2, C22.264(C2×C22⋊C4), SmallGroup(128,605)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.73D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >
Subgroups: 348 in 180 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C22⋊C8, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C22×C8, C23×C4, C22×Q8, C22.4Q16, C2×C22⋊C8, C2×Q8⋊C4, C2×C42⋊C2, C2×C22⋊Q8, C24.73D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4○D8, C8.C22, C23.23D4, C23.24D4, C23.38D4, D4.7D4, C23.20D4, C24.73D4
►On 64 points
(1 19)(2 26)(3 21)(4 28)(5 23)(6 30)(7 17)(8 32)(9 45)(10 40)(11 47)(12 34)(13 41)(14 36)(15 43)(16 38)(18 54)(20 56)(22 50)(24 52)(25 55)(27 49)(29 51)(31 53)(33 59)(35 61)(37 63)(39 57)(42 62)(44 64)(46 58)(48 60)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(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 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(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 14 5 10)(2 45 6 41)(3 12 7 16)(4 43 8 47)(9 20 13 24)(11 18 15 22)(17 48 21 44)(19 46 23 42)(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)| (1,19)(2,26)(3,21)(4,28)(5,23)(6,30)(7,17)(8,32)(9,45)(10,40)(11,47)(12,34)(13,41)(14,36)(15,43)(16,38)(18,54)(20,56)(22,50)(24,52)(25,55)(27,49)(29,51)(31,53)(33,59)(35,61)(37,63)(39,57)(42,62)(44,64)(46,58)(48,60), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(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), (1,14,5,10)(2,45,6,41)(3,12,7,16)(4,43,8,47)(9,20,13,24)(11,18,15,22)(17,48,21,44)(19,46,23,42)(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( (1,19)(2,26)(3,21)(4,28)(5,23)(6,30)(7,17)(8,32)(9,45)(10,40)(11,47)(12,34)(13,41)(14,36)(15,43)(16,38)(18,54)(20,56)(22,50)(24,52)(25,55)(27,49)(29,51)(31,53)(33,59)(35,61)(37,63)(39,57)(42,62)(44,64)(46,58)(48,60), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(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), (1,14,5,10)(2,45,6,41)(3,12,7,16)(4,43,8,47)(9,20,13,24)(11,18,15,22)(17,48,21,44)(19,46,23,42)(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([[(1,19),(2,26),(3,21),(4,28),(5,23),(6,30),(7,17),(8,32),(9,45),(10,40),(11,47),(12,34),(13,41),(14,36),(15,43),(16,38),(18,54),(20,56),(22,50),(24,52),(25,55),(27,49),(29,51),(31,53),(33,59),(35,61),(37,63),(39,57),(42,62),(44,64),(46,58),(48,60)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(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),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(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)], [(1,14,5,10),(2,45,6,41),(3,12,7,16),(4,43,8,47),(9,20,13,24),(11,18,15,22),(17,48,21,44),(19,46,23,42),(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 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D8 | C8.C22 |
| kernel | C24.73D4 | C22.4Q16 | C2×C22⋊C8 | C2×Q8⋊C4 | C2×C42⋊C2 | C2×C22⋊Q8 | C22⋊Q8 | C4⋊C4 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 8 | 2 |
Matrix representation of C24.73D4 ►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 | 1 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,0,1],[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],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[13,0,0,0,0,0,15,0,0,0,0,0,8,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,0,2,0,0,0,8,0,0,0,0,0,0,13,4,0,0,0,9,4] >;
C24.73D4 in GAP, Magma, Sage, TeX
C_2^4._{73}D_4 % in TeX
G:=Group("C2^4.73D4"); // GroupNames label
G:=SmallGroup(128,605);
// by ID
G=gap.SmallGroup(128,605);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,248,2028,124]);
// 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=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,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