direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23×SD16, C8⋊3C24, C4.2C25, Q8⋊1C24, D4.1C24, C24.196D4, (C23×C8)⋊13C2, (C2×C8)⋊16C23, (Q8×C23)⋊13C2, (C2×Q8)⋊19C23, C2.37(D4×C23), C4.28(C22×D4), (C2×C4).608C24, (C22×C8)⋊69C22, (D4×C23).21C2, (C22×C4).628D4, C23.894(C2×D4), (C2×D4).489C23, (C22×Q8)⋊67C22, (C23×C4).712C22, C22.165(C22×D4), (C22×C4).1590C23, (C22×D4).602C22, (C2×C4).881(C2×D4), SmallGroup(128,2307)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23×SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 1500 in 860 conjugacy classes, 476 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C22×C8, C2×SD16, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C22×Q8, C25, C23×C8, C22×SD16, D4×C23, Q8×C23, C23×SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C22×D4, C25, C22×SD16, D4×C23, C23×SD16
►On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(33 35)(34 38)(37 39)(41 45)(42 48)(44 46)(49 55)(51 53)(52 56)(57 61)(58 64)(60 62)
G:=sub<Sym(64)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(57,61)(58,64)(60,62)>;
G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,35)(34,38)(37,39)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(57,61)(58,64)(60,62) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,64),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,27),(26,30),(29,31),(33,35),(34,38),(37,39),(41,45),(42,48),(44,46),(49,55),(51,53),(52,56),(57,61),(58,64),(60,62)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 |
| kernel | C23×SD16 | C23×C8 | C22×SD16 | D4×C23 | Q8×C23 | C22×C4 | C24 | C23 |
| # reps | 1 | 1 | 28 | 1 | 1 | 7 | 1 | 16 |
Matrix representation of C23×SD16 ►in GL5(𝔽17)
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 5 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(5,GF(17))| [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],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,7,5,0,0,0,7,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16] >;
C23×SD16 in GAP, Magma, Sage, TeX
C_2^3\times {\rm SD}_{16} % in TeX
G:=Group("C2^3xSD16"); // GroupNames label
G:=SmallGroup(128,2307);
// by ID
G=gap.SmallGroup(128,2307);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations