direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.48D4, C24.182D4, C23.21Q16, C4⋊C4.51C23, C2.D8⋊54C22, C2.8(C22×Q16), (C2×C4).288C24, (C2×C8).142C23, (C22×C4).439D4, C23.664(C2×D4), (C2×Q8).65C23, C22.15(C2×Q16), Q8⋊C4⋊69C22, C22⋊C8.171C22, (C23×C4).558C22, (C22×C8).147C22, C22.548(C22×D4), C22⋊Q8.158C22, C22.122(C8⋊C22), (C22×C4).1005C23, C4.60(C22.D4), (C22×Q8).291C22, C22.111(C22.D4), (C2×C2.D8)⋊26C2, C4.98(C2×C4○D4), (C2×C4).848(C2×D4), C2.27(C2×C8⋊C22), (C2×Q8⋊C4)⋊24C2, (C22×C4⋊C4).46C2, (C2×C22⋊C8).31C2, (C2×C22⋊Q8).56C2, (C2×C4).846(C4○D4), (C2×C4⋊C4).925C22, C2.53(C2×C22.D4), SmallGroup(128,1822)
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], C2.D8 [×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×C2.D8 [×2], C23.48D4 [×8], C22×C4⋊C4, C2×C22⋊Q8, C2×C23.48D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×Q16 [×6], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C23.48D4 [×4], C2×C22.D4, C22×Q16, C2×C8⋊C22, C2×C23.48D4
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=cde3 >
►On 64 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(2 23)(4 17)(6 19)(8 21)(10 32)(12 26)(14 28)(16 30)(33 46)(35 48)(37 42)(39 44)(49 62)(51 64)(53 58)(55 60)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)(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 10 5 14)(2 31 6 27)(3 16 7 12)(4 29 8 25)(9 19 13 23)(11 17 15 21)(18 28 22 32)(20 26 24 30)(33 63 37 59)(34 49 38 53)(35 61 39 57)(36 55 40 51)(41 60 45 64)(42 54 46 50)(43 58 47 62)(44 52 48 56)
G:=sub<Sym(64)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (2,23)(4,17)(6,19)(8,21)(10,32)(12,26)(14,28)(16,30)(33,46)(35,48)(37,42)(39,44)(49,62)(51,64)(53,58)(55,60), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45)(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), (1,10,5,14)(2,31,6,27)(3,16,7,12)(4,29,8,25)(9,19,13,23)(11,17,15,21)(18,28,22,32)(20,26,24,30)(33,63,37,59)(34,49,38,53)(35,61,39,57)(36,55,40,51)(41,60,45,64)(42,54,46,50)(43,58,47,62)(44,52,48,56)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (2,23)(4,17)(6,19)(8,21)(10,32)(12,26)(14,28)(16,30)(33,46)(35,48)(37,42)(39,44)(49,62)(51,64)(53,58)(55,60), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45)(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), (1,10,5,14)(2,31,6,27)(3,16,7,12)(4,29,8,25)(9,19,13,23)(11,17,15,21)(18,28,22,32)(20,26,24,30)(33,63,37,59)(34,49,38,53)(35,61,39,57)(36,55,40,51)(41,60,45,64)(42,54,46,50)(43,58,47,62)(44,52,48,56) );
G=PermutationGroup([(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(2,23),(4,17),(6,19),(8,21),(10,32),(12,26),(14,28),(16,30),(33,46),(35,48),(37,42),(39,44),(49,62),(51,64),(53,58),(55,60)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45),(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)], [(1,10,5,14),(2,31,6,27),(3,16,7,12),(4,29,8,25),(9,19,13,23),(11,17,15,21),(18,28,22,32),(20,26,24,30),(33,63,37,59),(34,49,38,53),(35,61,39,57),(36,55,40,51),(41,60,45,64),(42,54,46,50),(43,58,47,62),(44,52,48,56)])
Matrix representation ►G ⊆ 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 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 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 10 |
| 0 | 0 | 0 | 0 | 10 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,10,0,0,0,0,10,1] >;
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 | Q16 | C8⋊C22 |
| kernel | C2×C23.48D4 | C2×C22⋊C8 | C2×Q8⋊C4 | C2×C2.D8 | C23.48D4 | 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._{48}D_4 % in TeX
G:=Group("C2xC2^3.48D4"); // GroupNames label
G:=SmallGroup(128,1822);
// by ID
G=gap.SmallGroup(128,1822);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,436,4037,1027,124]);
// 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*d*e^3>;
// generators/relations