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×C8).590C23, (C2×C4).194C24, (C22×C4).825D4, C22.116(C4×D4), C23.841(C2×D4), 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, (C2×C42).1112C22, (C22×C4).1510C23, (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, C4.2(C2×C4○D4), C2.4(C2×C4○D8), 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
Generators and relations for C2×C4×SD16
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 476 in 276 conjugacy classes, 156 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C22×C8, C2×SD16, C23×C4, C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C4×SD16, C2×C4×D4, C2×C4×Q8, C22×SD16, C2×C4×SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, C22×C4, C2×D4, C4○D4, C24, C4×D4, C2×SD16, C4○D8, C23×C4, C22×D4, C2×C4○D4, C4×SD16, C2×C4×D4, C22×SD16, C2×C4○D8, C2×C4×SD16
►On 64 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 33 55 47)(2 34 56 48)(3 35 49 41)(4 36 50 42)(5 37 51 43)(6 38 52 44)(7 39 53 45)(8 40 54 46)(9 24 63 26)(10 17 64 27)(11 18 57 28)(12 19 58 29)(13 20 59 30)(14 21 60 31)(15 22 61 32)(16 23 62 25)
(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 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(17 54)(18 49)(19 52)(20 55)(21 50)(22 53)(23 56)(24 51)(41 57)(42 60)(43 63)(44 58)(45 61)(46 64)(47 59)(48 62)
G:=sub<Sym(64)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,33,55,47)(2,34,56,48)(3,35,49,41)(4,36,50,42)(5,37,51,43)(6,38,52,44)(7,39,53,45)(8,40,54,46)(9,24,63,26)(10,17,64,27)(11,18,57,28)(12,19,58,29)(13,20,59,30)(14,21,60,31)(15,22,61,32)(16,23,62,25), (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)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,54)(18,49)(19,52)(20,55)(21,50)(22,53)(23,56)(24,51)(41,57)(42,60)(43,63)(44,58)(45,61)(46,64)(47,59)(48,62)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,33,55,47)(2,34,56,48)(3,35,49,41)(4,36,50,42)(5,37,51,43)(6,38,52,44)(7,39,53,45)(8,40,54,46)(9,24,63,26)(10,17,64,27)(11,18,57,28)(12,19,58,29)(13,20,59,30)(14,21,60,31)(15,22,61,32)(16,23,62,25), (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)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,54)(18,49)(19,52)(20,55)(21,50)(22,53)(23,56)(24,51)(41,57)(42,60)(43,63)(44,58)(45,61)(46,64)(47,59)(48,62) );
G=PermutationGroup([[(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,56),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(1,33,55,47),(2,34,56,48),(3,35,49,41),(4,36,50,42),(5,37,51,43),(6,38,52,44),(7,39,53,45),(8,40,54,46),(9,24,63,26),(10,17,64,27),(11,18,57,28),(12,19,58,29),(13,20,59,30),(14,21,60,31),(15,22,61,32),(16,23,62,25)], [(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),(9,37),(10,40),(11,35),(12,38),(13,33),(14,36),(15,39),(16,34),(17,54),(18,49),(19,52),(20,55),(21,50),(22,53),(23,56),(24,51),(41,57),(42,60),(43,63),(44,58),(45,61),(46,64),(47,59),(48,62)]])
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 |
Matrix representation of C2×C4×SD16 ►in 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] >;
C2×C4×SD16 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm 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