p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q8○SD32, D4○SD32, D4.13D8, Q8.13D8, D16⋊6C22, C16.4C23, C8.17C24, Q32⋊5C22, D8.6C23, SD32⋊5C22, Q16.6C23, M4(2).22D4, M5(2)⋊9C22, D4○D8⋊6C2, Q8○D8⋊5C2, D4○C16⋊4C2, C4○D16⋊5C2, C8.16(C2×D4), C4.50(C2×D8), C16⋊C22⋊6C2, (C2×C16)⋊6C22, (C2×SD32)⋊6C2, C4○D4.36D4, Q32⋊C2⋊5C2, C4○D8⋊2C22, C22.7(C2×D8), C4.23(C22×D4), C2.32(C22×D8), (C2×C8).295C23, (C2×Q16)⋊34C22, C8○D4.13C22, (C2×D8).98C22, (C2×C4).185(C2×D4), SmallGroup(128,2148)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 416 in 180 conjugacy classes, 90 normal (15 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8, D8 [×3], D8 [×3], SD16 [×6], Q16, Q16 [×3], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], C2×C16 [×3], M5(2) [×3], D16 [×3], SD32, SD32 [×9], Q32 [×3], C8○D4, C2×D8 [×3], C2×Q16 [×3], C4○D8 [×6], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), D4○C16, C2×SD32 [×3], C4○D16 [×3], C16⋊C22 [×3], Q32⋊C2 [×3], D4○D8, Q8○D8, Q8○SD32
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, Q8○SD32
Generators and relations
G = < a,b,c,d | a4=d2=1, b2=c8=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
►On 32 points
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 21 9 29)(2 22 10 30)(3 23 11 31)(4 24 12 32)(5 25 13 17)(6 26 14 18)(7 27 15 19)(8 28 16 20)
(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)
(1 22)(2 29)(3 20)(4 27)(5 18)(6 25)(7 32)(8 23)(9 30)(10 21)(11 28)(12 19)(13 26)(14 17)(15 24)(16 31)
G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,21,9,29)(2,22,10,30)(3,23,11,31)(4,24,12,32)(5,25,13,17)(6,26,14,18)(7,27,15,19)(8,28,16,20), (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), (1,22)(2,29)(3,20)(4,27)(5,18)(6,25)(7,32)(8,23)(9,30)(10,21)(11,28)(12,19)(13,26)(14,17)(15,24)(16,31)>;
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,21,9,29)(2,22,10,30)(3,23,11,31)(4,24,12,32)(5,25,13,17)(6,26,14,18)(7,27,15,19)(8,28,16,20), (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), (1,22)(2,29)(3,20)(4,27)(5,18)(6,25)(7,32)(8,23)(9,30)(10,21)(11,28)(12,19)(13,26)(14,17)(15,24)(16,31) );
G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,21,9,29),(2,22,10,30),(3,23,11,31),(4,24,12,32),(5,25,13,17),(6,26,14,18),(7,27,15,19),(8,28,16,20)], [(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)], [(1,22),(2,29),(3,20),(4,27),(5,18),(6,25),(7,32),(8,23),(9,30),(10,21),(11,28),(12,19),(13,26),(14,17),(15,24),(16,31)])
Matrix representation ►G ⊆ GL4(𝔽7) generated by
| 0 | 6 | 5 | 1 |
| 3 | 0 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 1 | 6 | 3 | 1 |
| 4 | 2 | 2 | 1 |
| 1 | 4 | 1 | 3 |
| 0 | 4 | 5 | 5 |
| 2 | 4 | 1 | 1 |
| 4 | 0 | 5 | 1 |
| 1 | 4 | 2 | 1 |
| 1 | 6 | 1 | 5 |
| 5 | 5 | 1 | 0 |
| 4 | 5 | 6 | 6 |
| 4 | 3 | 2 | 2 |
| 2 | 1 | 1 | 0 |
| 5 | 6 | 5 | 6 |
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,1,0,2,2,4,4,4,2,1,5,1,1,3,5,1],[4,1,1,5,0,4,6,5,5,2,1,1,1,1,5,0],[4,4,2,5,5,3,1,6,6,2,1,5,6,2,0,6] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | 16E | ··· | 16J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 | 16 | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | Q8○SD32 |
| kernel | Q8○SD32 | D4○C16 | C2×SD32 | C4○D16 | C16⋊C22 | Q32⋊C2 | D4○D8 | Q8○D8 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 6 | 2 | 4 |
In GAP, Magma, Sage, TeX
Q_8\circ SD_{32} % in TeX
G:=Group("Q8oSD32"); // GroupNames label
G:=SmallGroup(128,2148);
// by ID
G=gap.SmallGroup(128,2148);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,521,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^8=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations