direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4×SD16, C42.350D4, C42.684C23, C4○(C4×SD16), C8⋊6(C22×C4), C4.43(C4×D4), (C4×C8)⋊76C22, Q8⋊1(C22×C4), C4.14(C23×C4), (C4×Q8)⋊74C22, D4.1(C22×C4), C4.Q8⋊73C22, C4⋊C4.354C23, (C2×C4).194C24, (C2×C8).590C23, C22.116(C4×D4), C23.841(C2×D4), (C22×C4).825D4, C2.4(C22×SD16), (C2×D4).365C23, (C4×D4).288C22, C22.84(C4○D8), (C2×Q8).337C23, C22.81(C2×SD16), Q8⋊C4⋊100C22, (C22×C8).511C22, (C22×SD16).15C2, C22.138(C22×D4), D4⋊C4.214C22, (C22×C4).1510C23, (C2×C42).1112C22, (C2×SD16).176C22, (C22×D4).556C22, (C22×Q8).459C22, (C2×C4×C8)⋊34C2, (C2×C4×Q8)⋊31C2, C4○2(C2×C4.Q8), C2.54(C2×C4×D4), (C2×C8)⋊31(C2×C4), (C2×C4)○(C4×SD16), (C2×C4×D4).73C2, C2.4(C2×C4○D8), C4.2(C2×C4○D4), C4○3(C2×D4⋊C4), C4○2(C2×Q8⋊C4), (C2×Q8)⋊27(C2×C4), (C2×C4.Q8)⋊39C2, (C2×C4)○3(C4.Q8), (C2×C4)○4(D4⋊C4), (C2×C4)○3(Q8⋊C4), (C2×Q8⋊C4)⋊60C2, (C2×D4).175(C2×C4), (C2×C4).1574(C2×D4), (C2×D4⋊C4).40C2, (C2×C4).686(C4○D4), (C2×C4⋊C4).907C22, (C2×C4).465(C22×C4), (C2×C4)○2(C2×C4.Q8), (C2×C4)○3(C2×D4⋊C4), (C2×C4)○2(C2×Q8⋊C4), SmallGroup(128,1669)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 476 in 276 conjugacy classes, 156 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C22 [×16], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×24], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], SD16 [×16], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C22×C8 [×2], C2×SD16 [×12], C23×C4, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C4×SD16 [×8], C2×C4×D4, C2×C4×Q8, C22×SD16, C2×C4×SD16
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], SD16 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×SD16 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×SD16 [×4], C2×C4×D4, C22×SD16, C2×C4○D8, C2×C4×SD16
Generators and relations
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
►On 64 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(1 33 55 9)(2 34 56 10)(3 35 49 11)(4 36 50 12)(5 37 51 13)(6 38 52 14)(7 39 53 15)(8 40 54 16)(17 46 58 29)(18 47 59 30)(19 48 60 31)(20 41 61 32)(21 42 62 25)(22 43 63 26)(23 44 64 27)(24 45 57 28)
(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 30)(2 25)(3 28)(4 31)(5 26)(6 29)(7 32)(8 27)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(41 53)(42 56)(43 51)(44 54)(45 49)(46 52)(47 55)(48 50)
G:=sub<Sym(64)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,33,55,9)(2,34,56,10)(3,35,49,11)(4,36,50,12)(5,37,51,13)(6,38,52,14)(7,39,53,15)(8,40,54,16)(17,46,58,29)(18,47,59,30)(19,48,60,31)(20,41,61,32)(21,42,62,25)(22,43,63,26)(23,44,64,27)(24,45,57,28), (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), (1,30)(2,25)(3,28)(4,31)(5,26)(6,29)(7,32)(8,27)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(41,53)(42,56)(43,51)(44,54)(45,49)(46,52)(47,55)(48,50)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,33,55,9)(2,34,56,10)(3,35,49,11)(4,36,50,12)(5,37,51,13)(6,38,52,14)(7,39,53,15)(8,40,54,16)(17,46,58,29)(18,47,59,30)(19,48,60,31)(20,41,61,32)(21,42,62,25)(22,43,63,26)(23,44,64,27)(24,45,57,28), (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), (1,30)(2,25)(3,28)(4,31)(5,26)(6,29)(7,32)(8,27)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(41,53)(42,56)(43,51)(44,54)(45,49)(46,52)(47,55)(48,50) );
G=PermutationGroup([(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)], [(1,33,55,9),(2,34,56,10),(3,35,49,11),(4,36,50,12),(5,37,51,13),(6,38,52,14),(7,39,53,15),(8,40,54,16),(17,46,58,29),(18,47,59,30),(19,48,60,31),(20,41,61,32),(21,42,62,25),(22,43,63,26),(23,44,64,27),(24,45,57,28)], [(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)], [(1,30),(2,25),(3,28),(4,31),(5,26),(6,29),(7,32),(8,27),(9,59),(10,62),(11,57),(12,60),(13,63),(14,58),(15,61),(16,64),(17,38),(18,33),(19,36),(20,39),(21,34),(22,37),(23,40),(24,35),(41,53),(42,56),(43,51),(44,54),(45,49),(46,52),(47,55),(48,50)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,12,5,0,0,12,12],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | C4○D4 | C4○D8 |
| kernel | C2×C4×SD16 | C2×C4×C8 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.Q8 | C4×SD16 | C2×C4×D4 | C2×C4×Q8 | C22×SD16 | C2×SD16 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 16 | 2 | 2 | 8 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times SD_{16} % in TeX
G:=Group("C2xC4xSD16"); // GroupNames label
G:=SmallGroup(128,1669);
// by ID
G=gap.SmallGroup(128,1669);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations