direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8×SD16, C82⋊10C2, C42.633C23, C8⋊5(C2×C8), C8○(Q8⋊C8), (C8×Q8)⋊1C2, Q8⋊1(C2×C8), C8○(C8⋊2C8), C8○(C4.Q8), C2.5(C8×D4), C8○2(D4⋊C8), Q8⋊C8⋊44C2, C8⋊2C8⋊32C2, (C8×D4).2C2, D4.1(C2×C8), D4⋊C8.17C2, C8○(Q8⋊C4), C2.2(C8○D8), C4.8(C8○D4), (C2×C8).218D4, C4.8(C22×C8), C8○2(D4⋊C4), C4.Q8.12C4, C2.1(C4×SD16), (C2×SD16).8C4, C22.78(C4×D4), D4⋊C4.14C4, C4.126(C4○D8), C4⋊C8.271C22, (C4×C8).367C22, Q8⋊C4.13C4, (C4×SD16).18C2, C4.100(C2×SD16), (C4×D4).269C22, (C4×Q8).255C22, (C2×C8)○(Q8⋊C8), (C2×C8)○(C4×SD16), (C2×C8)○(C8⋊2C8), C4⋊C4.127(C2×C4), (C2×C8).155(C2×C4), (C2×D4).149(C2×C4), (C2×C4).1469(C2×D4), (C2×Q8).131(C2×C4), (C2×C4).494(C4○D4), (C2×C4).325(C22×C4), 2-Sylow(CU(3,5)), SmallGroup(128,308)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×SD16
G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 152 in 90 conjugacy classes, 52 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C82, D4⋊C8, Q8⋊C8, C8⋊2C8, C8×D4, C4×SD16, C8×Q8, C8×SD16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, SD16, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×SD16, C4○D8, C8×D4, C4×SD16, C8○D8, C8×SD16
►On 64 points
(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 21 13 63 31 52 39 43)(2 22 14 64 32 53 40 44)(3 23 15 57 25 54 33 45)(4 24 16 58 26 55 34 46)(5 17 9 59 27 56 35 47)(6 18 10 60 28 49 36 48)(7 19 11 61 29 50 37 41)(8 20 12 62 30 51 38 42)
(1 5)(2 6)(3 7)(4 8)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 29)(26 30)(27 31)(28 32)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)
G:=sub<Sym(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), (1,21,13,63,31,52,39,43)(2,22,14,64,32,53,40,44)(3,23,15,57,25,54,33,45)(4,24,16,58,26,55,34,46)(5,17,9,59,27,56,35,47)(6,18,10,60,28,49,36,48)(7,19,11,61,29,50,37,41)(8,20,12,62,30,51,38,42), (1,5)(2,6)(3,7)(4,8)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)>;
G:=Group( (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), (1,21,13,63,31,52,39,43)(2,22,14,64,32,53,40,44)(3,23,15,57,25,54,33,45)(4,24,16,58,26,55,34,46)(5,17,9,59,27,56,35,47)(6,18,10,60,28,49,36,48)(7,19,11,61,29,50,37,41)(8,20,12,62,30,51,38,42), (1,5)(2,6)(3,7)(4,8)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53) );
G=PermutationGroup([[(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)], [(1,21,13,63,31,52,39,43),(2,22,14,64,32,53,40,44),(3,23,15,57,25,54,33,45),(4,24,16,58,26,55,34,46),(5,17,9,59,27,56,35,47),(6,18,10,60,28,49,36,48),(7,19,11,61,29,50,37,41),(8,20,12,62,30,51,38,42)], [(1,5),(2,6),(3,7),(4,8),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,29),(26,30),(27,31),(28,32),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8AB | 8AC | ··· | 8AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | SD16 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8×SD16 | C82 | D4⋊C8 | Q8⋊C8 | C8⋊2C8 | C8×D4 | C4×SD16 | C8×Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | SD16 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8×SD16 ►in GL3(𝔽17) generated by
| 2 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 5 | 5 |
| 0 | 12 | 5 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,5,12,0,5,5],[1,0,0,0,1,0,0,0,16] >;
C8×SD16 in GAP, Magma, Sage, TeX
C_8\times {\rm SD}_{16} % in TeX
G:=Group("C8xSD16"); // GroupNames label
G:=SmallGroup(128,308);
// by ID
G=gap.SmallGroup(128,308);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations