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
Subgroups: 1500 in 860 conjugacy classes, 476 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4, C4 [×7], C4 [×8], C22 [×35], C22 [×64], C8 [×8], C2×C4 [×28], C2×C4 [×28], D4 [×8], D4 [×28], Q8 [×8], Q8 [×28], C23 [×15], C23 [×84], C2×C8 [×28], SD16 [×64], C22×C4 [×14], C22×C4 [×14], C2×D4 [×28], C2×D4 [×42], C2×Q8 [×28], C2×Q8 [×42], C24, C24 [×22], C22×C8 [×14], C2×SD16 [×112], C23×C4, C23×C4, C22×D4 [×14], C22×D4 [×7], C22×Q8 [×14], C22×Q8 [×7], C25, C23×C8, C22×SD16 [×28], D4×C23, Q8×C23, C23×SD16
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], SD16 [×8], C2×D4 [×28], C24 [×31], C2×SD16 [×28], C22×D4 [×14], C25, C22×SD16 [×14], D4×C23, C23×SD16
Generators and relations
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 >
►On 64 points
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(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 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 31)(27 29)(28 32)(33 39)(35 37)(36 40)(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)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(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)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(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)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(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)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(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),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(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),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45)], [(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),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,45),(42,48),(44,46),(49,55),(51,53),(52,56),(57,61),(58,64),(60,62)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2^3\times 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