p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊15SD16, Q8⋊1M4(2), C42.640C23, Q8⋊C8⋊38C2, D4⋊C8.2C2, (C8×Q8)⋊25C2, C8⋊C8⋊18C2, C8⋊2C8⋊29C2, C4.Q8.8C4, (C2×C8).379D4, C2.6(C4×SD16), D4⋊C4.7C4, C4.32(C8○D4), C8⋊6D4.11C2, Q8⋊C4.7C4, (C2×SD16).4C4, C4.130(C4○D8), C2.8(C8.26D4), C2.10(C8⋊9D4), C4⋊C8.222C22, (C4×C8).239C22, (C4×SD16).10C2, C4.102(C2×SD16), (C4×D4).10C22, C22.131(C4×D4), C4.26(C2×M4(2)), (C4×Q8).259C22, C4⋊C4.134(C2×C4), (C2×C8).103(C2×C4), (C2×D4).56(C2×C4), (C2×C4).1476(C2×D4), (C2×Q8).135(C2×C4), (C2×C4).501(C4○D4), (C2×C4).332(C22×C4), SmallGroup(128,315)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊15SD16
G = < a,b,c | a8=b8=c2=1, bab-1=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, 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, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C8⋊2C8, C8⋊6D4, C4×SD16, C8×Q8, C8⋊15SD16
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⋊9D4, C4×SD16, C8.26D4, C8⋊15SD16
►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 23 37 63 13 44 31 54)(2 20 38 60 14 41 32 51)(3 17 39 57 15 46 25 56)(4 22 40 62 16 43 26 53)(5 19 33 59 9 48 27 50)(6 24 34 64 10 45 28 55)(7 21 35 61 11 42 29 52)(8 18 36 58 12 47 30 49)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(26 30)(28 32)(34 38)(36 40)(41 64)(42 61)(43 58)(44 63)(45 60)(46 57)(47 62)(48 59)
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), (1,23,37,63,13,44,31,54)(2,20,38,60,14,41,32,51)(3,17,39,57,15,46,25,56)(4,22,40,62,16,43,26,53)(5,19,33,59,9,48,27,50)(6,24,34,64,10,45,28,55)(7,21,35,61,11,42,29,52)(8,18,36,58,12,47,30,49), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,56)(18,53)(19,50)(20,55)(21,52)(22,49)(23,54)(24,51)(26,30)(28,32)(34,38)(36,40)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59)>;
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), (1,23,37,63,13,44,31,54)(2,20,38,60,14,41,32,51)(3,17,39,57,15,46,25,56)(4,22,40,62,16,43,26,53)(5,19,33,59,9,48,27,50)(6,24,34,64,10,45,28,55)(7,21,35,61,11,42,29,52)(8,18,36,58,12,47,30,49), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,56)(18,53)(19,50)(20,55)(21,52)(22,49)(23,54)(24,51)(26,30)(28,32)(34,38)(36,40)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59) );
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)], [(1,23,37,63,13,44,31,54),(2,20,38,60,14,41,32,51),(3,17,39,57,15,46,25,56),(4,22,40,62,16,43,26,53),(5,19,33,59,9,48,27,50),(6,24,34,64,10,45,28,55),(7,21,35,61,11,42,29,52),(8,18,36,58,12,47,30,49)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16),(17,56),(18,53),(19,50),(20,55),(21,52),(22,49),(23,54),(24,51),(26,30),(28,32),(34,38),(36,40),(41,64),(42,61),(43,58),(44,63),(45,60),(46,57),(47,62),(48,59)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | SD16 | C4○D4 | M4(2) | C8○D4 | C4○D8 | C8.26D4 |
| kernel | C8⋊15SD16 | C8⋊C8 | D4⋊C8 | Q8⋊C8 | C8⋊2C8 | C8⋊6D4 | C4×SD16 | C8×Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | C2×C8 | C8 | C2×C4 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C8⋊15SD16 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 14 | 15 |
| 0 | 0 | 15 | 3 |
| 12 | 12 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 3 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 14 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,14,15,0,0,15,3],[12,5,0,0,12,12,0,0,0,0,16,3,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,14,0,0,0,16] >;
C8⋊15SD16 in GAP, Magma, Sage, TeX
C_8\rtimes_{15}{\rm SD}_{16} % in TeX
G:=Group("C8:15SD16"); // GroupNames label
G:=SmallGroup(128,315);
// by ID
G=gap.SmallGroup(128,315);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^3>;
// generators/relations