direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4×C4○D4, C23.25C24, C22.57C25, C24.494C23, C42.556C23, C4○D42, C4○(D4×Q8), D42⋊26C2, D4○2(C4×D4), Q8○2(C4×D4), D4⋊12(C2×D4), (D4×Q8)⋊33C2, Q8⋊12(C2×D4), C4○(D4⋊6D4), C4○(D4⋊5D4), C4○(Q8⋊5D4), C4○(Q8⋊6D4), C4⋊Q8⋊83C22, D4⋊6D4⋊46C2, D4⋊5D4⋊42C2, Q8⋊5D4⋊36C2, Q8⋊6D4⋊32C2, (C4×Q8)⋊93C22, C2.24(D4×C23), (C4×D4)⋊105C22, C4⋊C4.469C23, C4⋊1D4⋊48C22, C4⋊D4⋊73C22, (C2×C4).599C24, (C23×C4)⋊36C22, (C2×C42)⋊52C22, C4.113(C22×D4), C22⋊Q8⋊86C22, C22≀C2⋊32C22, C22.8(C22×D4), (C2×D4).453C23, C4.4D4⋊72C22, (C22×D4)⋊63C22, C22⋊C4.13C23, (C2×Q8).431C23, (C22×Q8)⋊64C22, C22.19C24⋊17C2, C42⋊C2⋊93C22, C2.10(C2.C25), (C22×C4).1194C23, C22.26C24⋊29C2, C22.D4⋊42C22, (C2×C4)○D42, (C2×C4)○(D4×Q8), (C2×Q8)○(C4×D4), (C2×C4×D4)⋊83C2, C4⋊4(C2×C4○D4), C4⋊C4○2(C4○D4), (C2×C4)⋊19(C2×D4), (C4×C4○D4)⋊20C2, C22⋊4(C2×C4○D4), (C2×C4)○(D4⋊6D4), (C2×C4)○(D4⋊5D4), C22⋊C4○2(C4○D4), (C2×C4)○(Q8⋊6D4), (C2×C4⋊C4)⋊134C22, (C2×C4○D4)⋊74C22, (C22×C4○D4)⋊20C2, C2.29(C22×C4○D4), (C2×C22⋊C4)⋊87C22, C4⋊C4○(C2×C4○D4), (C4×D4)○(C2×C4○D4), (C2×D4)○(C2×C4○D4), C22⋊C4○(C2×C4○D4), SmallGroup(128,2200)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4×C4○D4
G = < a,b,c,d,e | a4=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
Subgroups: 1300 in 824 conjugacy classes, 438 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C22.19C24, C22.26C24, D42, D4⋊5D4, D4⋊6D4, Q8⋊5D4, D4×Q8, Q8⋊6D4, C22×C4○D4, D4×C4○D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C25, D4×C23, C22×C4○D4, C2.C25, D4×C4○D4
►On 32 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)
(2 4)(5 7)(9 11)(13 15)(17 19)(22 24)(26 28)(30 32)
(1 14 25 23)(2 15 26 24)(3 16 27 21)(4 13 28 22)(5 11 19 32)(6 12 20 29)(7 9 17 30)(8 10 18 31)
(1 23 25 14)(2 24 26 15)(3 21 27 16)(4 22 28 13)(5 11 19 32)(6 12 20 29)(7 9 17 30)(8 10 18 31)
(1 8)(2 5)(3 6)(4 7)(9 13)(10 14)(11 15)(12 16)(17 28)(18 25)(19 26)(20 27)(21 29)(22 30)(23 31)(24 32)
G:=sub<Sym(32)| (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), (2,4)(5,7)(9,11)(13,15)(17,19)(22,24)(26,28)(30,32), (1,14,25,23)(2,15,26,24)(3,16,27,21)(4,13,28,22)(5,11,19,32)(6,12,20,29)(7,9,17,30)(8,10,18,31), (1,23,25,14)(2,24,26,15)(3,21,27,16)(4,22,28,13)(5,11,19,32)(6,12,20,29)(7,9,17,30)(8,10,18,31), (1,8)(2,5)(3,6)(4,7)(9,13)(10,14)(11,15)(12,16)(17,28)(18,25)(19,26)(20,27)(21,29)(22,30)(23,31)(24,32)>;
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), (2,4)(5,7)(9,11)(13,15)(17,19)(22,24)(26,28)(30,32), (1,14,25,23)(2,15,26,24)(3,16,27,21)(4,13,28,22)(5,11,19,32)(6,12,20,29)(7,9,17,30)(8,10,18,31), (1,23,25,14)(2,24,26,15)(3,21,27,16)(4,22,28,13)(5,11,19,32)(6,12,20,29)(7,9,17,30)(8,10,18,31), (1,8)(2,5)(3,6)(4,7)(9,13)(10,14)(11,15)(12,16)(17,28)(18,25)(19,26)(20,27)(21,29)(22,30)(23,31)(24,32) );
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)], [(2,4),(5,7),(9,11),(13,15),(17,19),(22,24),(26,28),(30,32)], [(1,14,25,23),(2,15,26,24),(3,16,27,21),(4,13,28,22),(5,11,19,32),(6,12,20,29),(7,9,17,30),(8,10,18,31)], [(1,23,25,14),(2,24,26,15),(3,21,27,16),(4,22,28,13),(5,11,19,32),(6,12,20,29),(7,9,17,30),(8,10,18,31)], [(1,8),(2,5),(3,6),(4,7),(9,13),(10,14),(11,15),(12,16),(17,28),(18,25),(19,26),(20,27),(21,29),(22,30),(23,31),(24,32)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2S | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 4S | ··· | 4AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C2.C25 |
| kernel | D4×C4○D4 | C2×C4×D4 | C4×C4○D4 | C22.19C24 | C22.26C24 | D42 | D4⋊5D4 | D4⋊6D4 | Q8⋊5D4 | D4×Q8 | Q8⋊6D4 | C22×C4○D4 | C4○D4 | D4 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 3 | 6 | 3 | 2 | 1 | 1 | 2 | 8 | 8 | 2 |
Matrix representation of D4×C4○D4 ►in GL4(𝔽5) generated by
| 1 | 2 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(5))| [1,4,0,0,2,4,0,0,0,0,1,0,0,0,0,1],[1,4,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,0,2,0,0,3,0] >;
D4×C4○D4 in GAP, Magma, Sage, TeX
D_4\times C_4\circ D_4
% in TeX
G:=Group("D4xC4oD4"); // GroupNames label
G:=SmallGroup(128,2200);
// by ID
G=gap.SmallGroup(128,2200);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,102]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^4=e^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations