p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).103D4, (C2×D4).25Q8, (C2×D4).201D4, C23.2(C2×Q8), (C22×C8).10C4, (C2×Q8).159D4, C8.46(C22⋊C4), C23.10(C4⋊C4), C4.10C42⋊7C2, C4.122(C4⋊D4), (C2×M4(2)).13C4, C4.84(C42⋊C2), (C22×C8).215C22, (C22×C4).663C23, C22.12(C22⋊Q8), C2.22(C23.7Q8), (C2×M4(2)).156C22, M4(2).8C22.5C2, (C2×C8).12(C2×C4), (C2×C8○D4).2C2, (C2×C8.C4)⋊3C2, (C2×C4).12(C4⋊C4), (C2×C4).230(C2×D4), C22.21(C2×C4⋊C4), C4.92(C2×C22⋊C4), (C22×C4).75(C2×C4), (C2×C4).738(C4○D4), (C2×C4).533(C22×C4), (C2×C4○D4).257C22, SmallGroup(128,545)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).103D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab2, ab=ba, ac=ca, dad-1=ab4, cbc-1=dbd-1=b-1, dcd-1=ab6c3 >
Subgroups: 212 in 126 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C4.10C42, M4(2).8C22, C2×C8.C4, C2×C8○D4, (C2×C8).103D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, (C2×C8).103D4
►On 32 points
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(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 15 22 28 5 11 18 32)(2 14 23 27 6 10 19 31)(3 13 24 26 7 9 20 30)(4 12 17 25 8 16 21 29)
(1 11 22 28 5 15 18 32)(2 10 23 27 6 14 19 31)(3 9 24 26 7 13 20 30)(4 16 17 25 8 12 21 29)
G:=sub<Sym(32)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,15,22,28,5,11,18,32)(2,14,23,27,6,10,19,31)(3,13,24,26,7,9,20,30)(4,12,17,25,8,16,21,29), (1,11,22,28,5,15,18,32)(2,10,23,27,6,14,19,31)(3,9,24,26,7,13,20,30)(4,16,17,25,8,12,21,29)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,15,22,28,5,11,18,32)(2,14,23,27,6,10,19,31)(3,13,24,26,7,9,20,30)(4,12,17,25,8,16,21,29), (1,11,22,28,5,15,18,32)(2,10,23,27,6,14,19,31)(3,9,24,26,7,13,20,30)(4,16,17,25,8,12,21,29) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31)], [(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)], [(1,15,22,28,5,11,18,32),(2,14,23,27,6,10,19,31),(3,13,24,26,7,9,20,30),(4,12,17,25,8,16,21,29)], [(1,11,22,28,5,15,18,32),(2,10,23,27,6,14,19,31),(3,9,24,26,7,13,20,30),(4,16,17,25,8,12,21,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | ··· | 8R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | D4 | C4○D4 | (C2×C8).103D4 |
| kernel | (C2×C8).103D4 | C4.10C42 | M4(2).8C22 | C2×C8.C4 | C2×C8○D4 | C22×C8 | C2×M4(2) | C2×C8 | C2×D4 | C2×D4 | C2×Q8 | C2×C4 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of (C2×C8).103D4 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 9 | 8 | 0 | 4 |
| 9 | 9 | 13 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 14 | 0 | 9 | 0 |
| 3 | 0 | 0 | 9 |
| 2 | 0 | 1 | 0 |
| 2 | 0 | 0 | 16 |
| 13 | 1 | 15 | 0 |
| 5 | 0 | 2 | 0 |
| 2 | 0 | 1 | 0 |
| 15 | 0 | 0 | 1 |
| 13 | 1 | 15 | 0 |
| 3 | 0 | 2 | 0 |
G:=sub<GL(4,GF(17))| [0,4,9,9,13,0,8,9,0,0,0,13,0,0,4,0],[2,0,14,3,0,2,0,0,0,0,9,0,0,0,0,9],[2,2,13,5,0,0,1,0,1,0,15,2,0,16,0,0],[2,15,13,3,0,0,1,0,1,0,15,2,0,1,0,0] >;
(C2×C8).103D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{103}D_4 % in TeX
G:=Group("(C2xC8).103D4"); // GroupNames label
G:=SmallGroup(128,545);
// by ID
G=gap.SmallGroup(128,545);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,2804,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*b^6*c^3>;
// generators/relations