p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊9SD16, C82⋊13C2, C8⋊5M4(2), C42.647C23, Q8⋊C8⋊41C2, C8⋊2C8⋊30C2, D4⋊C8.15C2, C8⋊4Q8⋊30C2, (C2×C8).285D4, C4.Q8.11C4, C2.7(C4×SD16), D4⋊C4.2C4, C2.11(C8○D8), C4.14(C8○D4), C8⋊6D4.13C2, C2.7(C8⋊6D4), Q8⋊C4.1C4, (C2×SD16).7C4, C4.8(C2×M4(2)), C4.133(C4○D8), C4⋊C8.225C22, (C4×C8).425C22, (C4×SD16).11C2, C4.103(C2×SD16), (C4×D4).14C22, C22.138(C4×D4), (C4×Q8).13C22, C4⋊C4.60(C2×C4), (C2×C8).182(C2×C4), (C2×D4).59(C2×C4), (C2×Q8).53(C2×C4), (C2×C4).1483(C2×D4), (C2×C4).508(C4○D4), (C2×C4).339(C22×C4), SmallGroup(128,322)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊9SD16
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5, cbc=b3 >
Subgroups: 152 in 83 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C82, D4⋊C8, Q8⋊C8, C8⋊2C8, C8⋊6D4, C4×SD16, C8⋊4Q8, C8⋊9SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), SD16, C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×SD16, C4○D8, C8⋊6D4, C4×SD16, C8○D8, C8⋊9SD16
►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 9 59 31 47 34 55)(2 22 10 60 32 48 35 56)(3 23 11 61 25 41 36 49)(4 24 12 62 26 42 37 50)(5 17 13 63 27 43 38 51)(6 18 14 64 28 44 39 52)(7 19 15 57 29 45 40 53)(8 20 16 58 30 46 33 54)
(2 6)(4 8)(9 34)(10 39)(11 36)(12 33)(13 38)(14 35)(15 40)(16 37)(17 63)(18 60)(19 57)(20 62)(21 59)(22 64)(23 61)(24 58)(26 30)(28 32)(41 49)(42 54)(43 51)(44 56)(45 53)(46 50)(47 55)(48 52)
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), (1,21,9,59,31,47,34,55)(2,22,10,60,32,48,35,56)(3,23,11,61,25,41,36,49)(4,24,12,62,26,42,37,50)(5,17,13,63,27,43,38,51)(6,18,14,64,28,44,39,52)(7,19,15,57,29,45,40,53)(8,20,16,58,30,46,33,54), (2,6)(4,8)(9,34)(10,39)(11,36)(12,33)(13,38)(14,35)(15,40)(16,37)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(26,30)(28,32)(41,49)(42,54)(43,51)(44,56)(45,53)(46,50)(47,55)(48,52)>;
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), (1,21,9,59,31,47,34,55)(2,22,10,60,32,48,35,56)(3,23,11,61,25,41,36,49)(4,24,12,62,26,42,37,50)(5,17,13,63,27,43,38,51)(6,18,14,64,28,44,39,52)(7,19,15,57,29,45,40,53)(8,20,16,58,30,46,33,54), (2,6)(4,8)(9,34)(10,39)(11,36)(12,33)(13,38)(14,35)(15,40)(16,37)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(26,30)(28,32)(41,49)(42,54)(43,51)(44,56)(45,53)(46,50)(47,55)(48,52) );
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)], [(1,21,9,59,31,47,34,55),(2,22,10,60,32,48,35,56),(3,23,11,61,25,41,36,49),(4,24,12,62,26,42,37,50),(5,17,13,63,27,43,38,51),(6,18,14,64,28,44,39,52),(7,19,15,57,29,45,40,53),(8,20,16,58,30,46,33,54)], [(2,6),(4,8),(9,34),(10,39),(11,36),(12,33),(13,38),(14,35),(15,40),(16,37),(17,63),(18,60),(19,57),(20,62),(21,59),(22,64),(23,61),(24,58),(26,30),(28,32),(41,49),(42,54),(43,51),(44,56),(45,53),(46,50),(47,55),(48,52)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8X | 8Y | 8Z | 8AA | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | M4(2) | SD16 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8⋊9SD16 | C82 | D4⋊C8 | Q8⋊C8 | C8⋊2C8 | C8⋊6D4 | C4×SD16 | C8⋊4Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8⋊9SD16 ►in GL4(𝔽17) generated by
| 2 | 4 | 0 | 0 |
| 15 | 15 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 13 | 15 |
| 10 | 10 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 16 |
G:=sub<GL(4,GF(17))| [2,15,0,0,4,15,0,0,0,0,2,13,0,0,2,15],[10,12,0,0,10,0,0,0,0,0,16,0,0,0,0,16],[1,16,0,0,0,16,0,0,0,0,1,15,0,0,0,16] >;
C8⋊9SD16 in GAP, Magma, Sage, TeX
C_8\rtimes_9{\rm SD}_{16} % in TeX
G:=Group("C8:9SD16"); // GroupNames label
G:=SmallGroup(128,322);
// by ID
G=gap.SmallGroup(128,322);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^3>;
// generators/relations