direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.47D4, C24.180D4, C23.39SD16, C4⋊C4.49C23, C4.Q8⋊60C22, (C2×C4).284C24, (C2×C8).309C23, C23.662(C2×D4), (C22×C4).435D4, (C2×Q8).63C23, Q8⋊C4⋊80C22, C22.23(C2×SD16), C2.12(C22×SD16), C22⋊C8.213C22, (C22×C8).346C22, (C23×C4).554C22, C22.544(C22×D4), C22⋊Q8.156C22, (C22×C4).1003C23, C4.56(C22.D4), (C22×Q8).289C22, C22.110(C8.C22), C22.107(C22.D4), (C2×C4.Q8)⋊31C2, C4.94(C2×C4○D4), (C2×C4).846(C2×D4), (C2×Q8⋊C4)⋊39C2, (C22×C4⋊C4).45C2, (C2×C22⋊C8).39C2, C2.26(C2×C8.C22), (C2×C22⋊Q8).54C2, (C2×C4).842(C4○D4), (C2×C4⋊C4).924C22, C2.49(C2×C22.D4), SmallGroup(128,1818)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 412 in 230 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×34], Q8 [×6], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×16], C2×Q8 [×2], C2×Q8 [×5], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×Q8, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C4.Q8 [×2], C23.47D4 [×8], C22×C4⋊C4, C2×C22⋊Q8, C2×C23.47D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×SD16 [×6], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C23.47D4 [×4], C2×C22.D4, C22×SD16, C2×C8.C22, C2×C23.47D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
►On 64 points
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(2 22)(4 24)(6 18)(8 20)(10 29)(12 31)(14 25)(16 27)(33 42)(35 44)(37 46)(39 48)(50 64)(52 58)(54 60)(56 62)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(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 28 6 32)(3 12 7 16)(4 26 8 30)(9 18 13 22)(11 24 15 20)(17 29 21 25)(19 27 23 31)(33 61 37 57)(34 50 38 54)(35 59 39 63)(36 56 40 52)(41 58 45 62)(42 55 46 51)(43 64 47 60)(44 53 48 49)
G:=sub<Sym(64)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (2,22)(4,24)(6,18)(8,20)(10,29)(12,31)(14,25)(16,27)(33,42)(35,44)(37,46)(39,48)(50,64)(52,58)(54,60)(56,62), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62), (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)(2,28,6,32)(3,12,7,16)(4,26,8,30)(9,18,13,22)(11,24,15,20)(17,29,21,25)(19,27,23,31)(33,61,37,57)(34,50,38,54)(35,59,39,63)(36,56,40,52)(41,58,45,62)(42,55,46,51)(43,64,47,60)(44,53,48,49)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (2,22)(4,24)(6,18)(8,20)(10,29)(12,31)(14,25)(16,27)(33,42)(35,44)(37,46)(39,48)(50,64)(52,58)(54,60)(56,62), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,63)(50,64)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62), (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)(2,28,6,32)(3,12,7,16)(4,26,8,30)(9,18,13,22)(11,24,15,20)(17,29,21,25)(19,27,23,31)(33,61,37,57)(34,50,38,54)(35,59,39,63)(36,56,40,52)(41,58,45,62)(42,55,46,51)(43,64,47,60)(44,53,48,49) );
G=PermutationGroup([(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(2,22),(4,24),(6,18),(8,20),(10,29),(12,31),(14,25),(16,27),(33,42),(35,44),(37,46),(39,48),(50,64),(52,58),(54,60),(56,62)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41),(49,63),(50,64),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62)], [(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),(2,28,6,32),(3,12,7,16),(4,26,8,30),(9,18,13,22),(11,24,15,20),(17,29,21,25),(19,27,23,31),(33,61,37,57),(34,50,38,54),(35,59,39,63),(36,56,40,52),(41,58,45,62),(42,55,46,51),(43,64,47,60),(44,53,48,49)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 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 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 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 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 3 | 8 |
| 0 | 0 | 0 | 16 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 5 | 12 |
G:=sub<GL(5,GF(17))| [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],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,16],[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],[16,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,3,16,0,0,0,8,14],[1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,5,5,0,0,0,2,12] >;
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | SD16 | C8.C22 |
| kernel | C2×C23.47D4 | C2×C22⋊C8 | C2×Q8⋊C4 | C2×C4.Q8 | C23.47D4 | C22×C4⋊C4 | C2×C22⋊Q8 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{47}D_4 % in TeX
G:=Group("C2xC2^3.47D4"); // GroupNames label
G:=SmallGroup(128,1818);
// by ID
G=gap.SmallGroup(128,1818);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,4037,1027,124]);
// 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,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*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