p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.157D4, C23.18Q16, C23.42SD16, C22.4Q16⋊6C2, C23.748(C2×D4), (C22×C4).278D4, C22.26(C2×Q16), (C22×C8).17C22, C22.50(C2×SD16), C4.19(C42⋊C2), C22.63(C8⋊C22), (C23×C4).244C22, C23.7Q8.10C2, C23.200(C22⋊C4), (C22×C4).1335C23, C22.24(Q8⋊C4), C2.1(C23.48D4), C2.1(C23.46D4), C2.12(C23.34D4), C4.104(C22.D4), C2.19(C23.37D4), C22.79(C22.D4), (C2×C4⋊C4)⋊29C4, C4⋊C4.195(C2×C4), (C2×C4).1325(C2×D4), (C22×C4⋊C4).14C2, (C2×C4⋊C4).42C22, (C2×C22⋊C8).18C2, C2.19(C2×Q8⋊C4), (C2×C4).741(C4○D4), (C2×C4).368(C22×C4), (C22×C4).267(C2×C4), (C2×C4).127(C22⋊C4), C22.258(C2×C22⋊C4), SmallGroup(128,556)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.157D4
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=bce3 >
Subgroups: 348 in 172 conjugacy classes, 68 normal (18 characteristic)
C1, C2, 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, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C22×C4⋊C4, C24.157D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4○D4, Q8⋊C4, C2×C22⋊C4, C42⋊C2, C22.D4, C2×SD16, C2×Q16, C8⋊C22, C23.34D4, C2×Q8⋊C4, C23.37D4, C23.46D4, C23.48D4, C24.157D4
►On 64 points
Generators in S64
(1 54)(2 37)(3 56)(4 39)(5 50)(6 33)(7 52)(8 35)(9 22)(10 30)(11 24)(12 32)(13 18)(14 26)(15 20)(16 28)(17 44)(19 46)(21 48)(23 42)(25 45)(27 47)(29 41)(31 43)(34 59)(36 61)(38 63)(40 57)(49 64)(51 58)(53 60)(55 62)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 23 54 10)(2 45 55 25)(3 21 56 16)(4 43 49 31)(5 19 50 14)(6 41 51 29)(7 17 52 12)(8 47 53 27)(9 33 22 58)(11 39 24 64)(13 37 18 62)(15 35 20 60)(26 40 46 57)(28 38 48 63)(30 36 42 61)(32 34 44 59)
G:=sub<Sym(64)| (1,54)(2,37)(3,56)(4,39)(5,50)(6,33)(7,52)(8,35)(9,22)(10,30)(11,24)(12,32)(13,18)(14,26)(15,20)(16,28)(17,44)(19,46)(21,48)(23,42)(25,45)(27,47)(29,41)(31,43)(34,59)(36,61)(38,63)(40,57)(49,64)(51,58)(53,60)(55,62), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,23,54,10)(2,45,55,25)(3,21,56,16)(4,43,49,31)(5,19,50,14)(6,41,51,29)(7,17,52,12)(8,47,53,27)(9,33,22,58)(11,39,24,64)(13,37,18,62)(15,35,20,60)(26,40,46,57)(28,38,48,63)(30,36,42,61)(32,34,44,59)>;
G:=Group( (1,54)(2,37)(3,56)(4,39)(5,50)(6,33)(7,52)(8,35)(9,22)(10,30)(11,24)(12,32)(13,18)(14,26)(15,20)(16,28)(17,44)(19,46)(21,48)(23,42)(25,45)(27,47)(29,41)(31,43)(34,59)(36,61)(38,63)(40,57)(49,64)(51,58)(53,60)(55,62), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (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), (1,23,54,10)(2,45,55,25)(3,21,56,16)(4,43,49,31)(5,19,50,14)(6,41,51,29)(7,17,52,12)(8,47,53,27)(9,33,22,58)(11,39,24,64)(13,37,18,62)(15,35,20,60)(26,40,46,57)(28,38,48,63)(30,36,42,61)(32,34,44,59) );
G=PermutationGroup([[(1,54),(2,37),(3,56),(4,39),(5,50),(6,33),(7,52),(8,35),(9,22),(10,30),(11,24),(12,32),(13,18),(14,26),(15,20),(16,28),(17,44),(19,46),(21,48),(23,42),(25,45),(27,47),(29,41),(31,43),(34,59),(36,61),(38,63),(40,57),(49,64),(51,58),(53,60),(55,62)], [(1,61),(2,62),(3,63),(4,64),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(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)], [(1,23,54,10),(2,45,55,25),(3,21,56,16),(4,43,49,31),(5,19,50,14),(6,41,51,29),(7,17,52,12),(8,47,53,27),(9,33,22,58),(11,39,24,64),(13,37,18,62),(15,35,20,60),(26,40,46,57),(28,38,48,63),(30,36,42,61),(32,34,44,59)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | SD16 | Q16 | C8⋊C22 |
| kernel | C24.157D4 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C22×C4⋊C4 | C2×C4⋊C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 8 | 4 | 4 | 2 |
Matrix representation of C24.157D4 ►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 |
| 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 | 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 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 3 | 3 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 10 |
| 0 | 0 | 0 | 10 | 1 |
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],[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],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,3,3,0,0,0,14,3],[13,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,10,0,0,0,10,1] >;
C24.157D4 in GAP, Magma, Sage, TeX
C_2^4._{157}D_4 % in TeX
G:=Group("C2^4.157D4"); // GroupNames label
G:=SmallGroup(128,556);
// by ID
G=gap.SmallGroup(128,556);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,394,2804,718,172]);
// 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=b*c*e^3>;
// generators/relations