p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊11D4, C8⋊D4⋊3C2, C8⋊8D4⋊43C2, C8.119(C2×D4), (C2×D4).216D4, C4⋊C4.26C23, (C2×Q8).171D4, C23.75(C2×D4), C2.D8⋊70C22, C4.Q8⋊56C22, C22⋊Q8⋊3C22, C4.69(C4⋊D4), (C2×C4).261C24, (C2×C8).253C23, (C22×C8)⋊41C22, (C22×SD16)⋊2C2, (C2×D4).64C23, C4.155(C22×D4), (C2×Q8).52C23, C2.16(D4○SD16), Q8⋊C4⋊53C22, C4⋊D4.19C22, C23.38D4⋊34C2, C23.25D4⋊27C2, C23.37D4⋊34C2, C22.86(C4⋊D4), (C2×M4(2))⋊53C22, (C22×C4).983C23, C22.521(C22×D4), D4⋊C4.129C22, C22.29C24.11C2, (C2×SD16).134C22, (C22×D4).349C22, (C22×Q8).282C22, C42⋊C2.110C22, C23.38C23⋊10C2, (C2×C8○D4)⋊1C2, C4.28(C2×C4○D4), (C2×C4).129(C2×D4), C2.79(C2×C4⋊D4), (C2×C4).282(C4○D4), (C2×C4○D4).301C22, SmallGroup(128,1789)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 508 in 246 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×16], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×9], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×SD16 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.37D4, C23.38D4, C23.25D4, C8⋊8D4 [×4], C8⋊D4 [×4], C22.29C24, C23.38C23, C2×C8○D4, C22×SD16, (C2×C8)⋊11D4
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○SD16 [×2], (C2×C8)⋊11D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=b3, dcd=c-1 >
►On 32 points
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 10 31 21)(2 13 32 24)(3 16 25 19)(4 11 26 22)(5 14 27 17)(6 9 28 20)(7 12 29 23)(8 15 30 18)
(2 4)(3 7)(6 8)(9 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)(25 29)(26 32)(28 30)
G:=sub<Sym(32)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,10,31,21)(2,13,32,24)(3,16,25,19)(4,11,26,22)(5,14,27,17)(6,9,28,20)(7,12,29,23)(8,15,30,18), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,29)(26,32)(28,30)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,10,31,21)(2,13,32,24)(3,16,25,19)(4,11,26,22)(5,14,27,17)(6,9,28,20)(7,12,29,23)(8,15,30,18), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,29)(26,32)(28,30) );
G=PermutationGroup([(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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)], [(1,10,31,21),(2,13,32,24),(3,16,25,19),(4,11,26,22),(5,14,27,17),(6,9,28,20),(7,12,29,23),(8,15,30,18)], [(2,4),(3,7),(6,8),(9,18),(10,21),(11,24),(12,19),(13,22),(14,17),(15,20),(16,23),(25,29),(26,32),(28,30)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 7 |
| 0 | 0 | 5 | 0 | 5 | 7 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 16 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,12,12,5,5,0,0,5,12,12,0,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,1,0,0,0,1,0,0,0,0,1,16,0,1,0,0,0,2,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D4○SD16 |
| kernel | (C2×C8)⋊11D4 | C23.37D4 | C23.38D4 | C23.25D4 | C8⋊8D4 | C8⋊D4 | C22.29C24 | C23.38C23 | C2×C8○D4 | C22×SD16 | C2×C8 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_{11}D_4 % in TeX
G:=Group("(C2xC8):11D4"); // GroupNames label
G:=SmallGroup(128,1789);
// by ID
G=gap.SmallGroup(128,1789);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,2804,172]);
// 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^3,d*c*d=c^-1>;
// generators/relations