p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.155D4, C23.17Q16, C23.41SD16, C4.53C22≀C2, Q8⋊3(C22⋊C4), (C2×Q8).206D4, (C22×Q8)⋊14C4, (Q8×C23).3C2, C2.1(Q8⋊D4), (C22×C4).263D4, C23.738(C2×D4), (C22×C8).6C22, C22.23(C2×Q16), C22.73C22≀C2, C22⋊3(Q8⋊C4), C2.1(C22⋊Q16), C22.45(C2×SD16), C2.12(C24⋊3C4), C23.7Q8.5C2, (C23×C4).232C22, C23.195(C22⋊C4), (C22×C4).1321C23, C22.46(C8.C22), (C22×Q8).380C22, C2.17(C23.38D4), C4.2(C2×C22⋊C4), (C2×Q8⋊C4)⋊1C2, (C2×C4).1311(C2×D4), (C2×C22⋊C8).12C2, (C2×C4⋊C4).29C22, (C2×Q8).183(C2×C4), C2.17(C2×Q8⋊C4), (C22×C4).261(C2×C4), (C2×C4).359(C22×C4), (C2×C4).120(C22⋊C4), C22.240(C2×C22⋊C4), SmallGroup(128,519)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.155D4
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=ce3 >
Subgroups: 540 in 296 conjugacy classes, 84 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C22⋊C8, Q8⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C22×Q8, C22×Q8, C23.7Q8, C2×C22⋊C8, C2×Q8⋊C4, Q8×C23, C24.155D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C22≀C2, C2×SD16, C2×Q16, C8.C22, C24⋊3C4, C2×Q8⋊C4, C23.38D4, Q8⋊D4, C22⋊Q16, C24.155D4
►On 64 points
Generators in S64
(1 35)(2 55)(3 37)(4 49)(5 39)(6 51)(7 33)(8 53)(9 47)(10 22)(11 41)(12 24)(13 43)(14 18)(15 45)(16 20)(17 26)(19 28)(21 30)(23 32)(25 42)(27 44)(29 46)(31 48)(34 58)(36 60)(38 62)(40 64)(50 63)(52 57)(54 59)(56 61)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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 18 35 27)(2 30 36 21)(3 24 37 25)(4 28 38 19)(5 22 39 31)(6 26 40 17)(7 20 33 29)(8 32 34 23)(9 55 47 60)(10 63 48 50)(11 53 41 58)(12 61 42 56)(13 51 43 64)(14 59 44 54)(15 49 45 62)(16 57 46 52)
G:=sub<Sym(64)| (1,35)(2,55)(3,37)(4,49)(5,39)(6,51)(7,33)(8,53)(9,47)(10,22)(11,41)(12,24)(13,43)(14,18)(15,45)(16,20)(17,26)(19,28)(21,30)(23,32)(25,42)(27,44)(29,46)(31,48)(34,58)(36,60)(38,62)(40,64)(50,63)(52,57)(54,59)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,18,35,27)(2,30,36,21)(3,24,37,25)(4,28,38,19)(5,22,39,31)(6,26,40,17)(7,20,33,29)(8,32,34,23)(9,55,47,60)(10,63,48,50)(11,53,41,58)(12,61,42,56)(13,51,43,64)(14,59,44,54)(15,49,45,62)(16,57,46,52)>;
G:=Group( (1,35)(2,55)(3,37)(4,49)(5,39)(6,51)(7,33)(8,53)(9,47)(10,22)(11,41)(12,24)(13,43)(14,18)(15,45)(16,20)(17,26)(19,28)(21,30)(23,32)(25,42)(27,44)(29,46)(31,48)(34,58)(36,60)(38,62)(40,64)(50,63)(52,57)(54,59)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,25)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,18,35,27)(2,30,36,21)(3,24,37,25)(4,28,38,19)(5,22,39,31)(6,26,40,17)(7,20,33,29)(8,32,34,23)(9,55,47,60)(10,63,48,50)(11,53,41,58)(12,61,42,56)(13,51,43,64)(14,59,44,54)(15,49,45,62)(16,57,46,52) );
G=PermutationGroup([[(1,35),(2,55),(3,37),(4,49),(5,39),(6,51),(7,33),(8,53),(9,47),(10,22),(11,41),(12,24),(13,43),(14,18),(15,45),(16,20),(17,26),(19,28),(21,30),(23,32),(25,42),(27,44),(29,46),(31,48),(34,58),(36,60),(38,62),(40,64),(50,63),(52,57),(54,59),(56,61)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,25),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(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)], [(1,18,35,27),(2,30,36,21),(3,24,37,25),(4,28,38,19),(5,22,39,31),(6,26,40,17),(7,20,33,29),(8,32,34,23),(9,55,47,60),(10,63,48,50),(11,53,41,58),(12,61,42,56),(13,51,43,64),(14,59,44,54),(15,49,45,62),(16,57,46,52)]])
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 | D4 | SD16 | Q16 | C8.C22 |
| kernel | C24.155D4 | C23.7Q8 | C2×C22⋊C8 | C2×Q8⋊C4 | Q8×C23 | C22×Q8 | C22×C4 | C2×Q8 | C24 | C23 | C23 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 3 | 8 | 1 | 4 | 4 | 2 |
Matrix representation of C24.155D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 15 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 15 |
| 0 | 0 | 0 | 0 | 8 | 7 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,10,7,0,0,0,0,15,7],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,8,0,0,0,0,15,7] >;
C24.155D4 in GAP, Magma, Sage, TeX
C_2^4._{155}D_4 % in TeX
G:=Group("C2^4.155D4"); // GroupNames label
G:=SmallGroup(128,519);
// by ID
G=gap.SmallGroup(128,519);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,1018,248]);
// 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=c*e^3>;
// generators/relations