p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊12D4, C8⋊2D4⋊3C2, C8⋊7D4⋊34C2, C8.109(C2×D4), (C2×D4).217D4, C2.10(D4○D8), (C22×D8)⋊16C2, C4⋊C4.27C23, C4⋊D4⋊3C22, (C2×Q8).172D4, C23.76(C2×D4), C2.D8⋊47C22, C4.Q8⋊51C22, C4.70(C4⋊D4), (C2×C8).254C23, (C2×C4).262C24, (C22×C8)⋊31C22, (C2×D4).65C23, C4.156(C22×D4), D4⋊C4⋊52C22, (C2×D8).119C22, C22.29C24⋊10C2, C23.37D4⋊35C2, C23.25D4⋊28C2, C22.87(C4⋊D4), (C2×M4(2))⋊54C22, (C22×C4).984C23, C22.522(C22×D4), (C22×D4).350C22, C42⋊C2.111C22, (C2×C8○D4)⋊2C2, C4.29(C2×C4○D4), (C2×C4).130(C2×D4), C2.80(C2×C4⋊D4), (C2×C4).283(C4○D4), (C2×C4○D4).302C22, SmallGroup(128,1790)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 620 in 262 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×28], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×26], Q8 [×2], C23, C23 [×2], C23 [×12], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], D8 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×6], C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], D4⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×4], C4⋊D4 [×8], C4.4D4 [×2], C4⋊1D4 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8 [×4], C2×D8 [×4], C22×D4 [×2], C2×C4○D4, C23.37D4 [×2], C23.25D4, C8⋊7D4 [×4], C8⋊2D4 [×4], C22.29C24 [×2], C2×C8○D4, C22×D8, (C2×C8)⋊12D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8 [×2], (C2×C8)⋊12D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
►On 32 points
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 9 27 23)(2 16 28 22)(3 15 29 21)(4 14 30 20)(5 13 31 19)(6 12 32 18)(7 11 25 17)(8 10 26 24)
(1 8)(2 7)(3 6)(4 5)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(25 28)(26 27)(29 32)(30 31)
G:=sub<Sym(32)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,9,27,23)(2,16,28,22)(3,15,29,21)(4,14,30,20)(5,13,31,19)(6,12,32,18)(7,11,25,17)(8,10,26,24), (1,8)(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(25,28)(26,27)(29,32)(30,31)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,9,27,23)(2,16,28,22)(3,15,29,21)(4,14,30,20)(5,13,31,19)(6,12,32,18)(7,11,25,17)(8,10,26,24), (1,8)(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(25,28)(26,27)(29,32)(30,31) );
G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)], [(1,9,27,23),(2,16,28,22),(3,15,29,21),(4,14,30,20),(5,13,31,19),(6,12,32,18),(7,11,25,17),(8,10,26,24)], [(1,8),(2,7),(3,6),(4,5),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(25,28),(26,27),(29,32),(30,31)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,14,14,0,0,0,0,14,3,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | D4 | C4○D4 | D4○D8 |
| kernel | (C2×C8)⋊12D4 | C23.37D4 | C23.25D4 | C8⋊7D4 | C8⋊2D4 | C22.29C24 | C2×C8○D4 | C22×D8 | C2×C8 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_{12}D_4 % in TeX
G:=Group("(C2xC8):12D4"); // GroupNames label
G:=SmallGroup(128,1790);
// by ID
G=gap.SmallGroup(128,1790);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations