p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8○D4⋊1C4, D4.2(C4⋊C4), C4○D4.23D4, (C2×C8).104D4, Q8.2(C4⋊C4), (C2×D4).26Q8, (C2×Q8).19Q8, C8.47(C22⋊C4), C23.4(C4○D4), C4.22(C22⋊Q8), C4.184(C4⋊D4), M4(2).30(C2×C4), M4(2)⋊4C4⋊12C2, C23.25D4⋊16C2, C22.40(C4⋊D4), (C22×C8).216C22, C42⋊C2.3C22, C42⋊C22.2C2, (C22×C4).664C23, C22.24(C22⋊Q8), C2.23(C23.7Q8), C22.14(C42⋊C2), (C2×M4(2)).157C22, C4.6(C2×C4⋊C4), (C2×C4).4(C2×Q8), (C2×C8).60(C2×C4), (C2×C8○D4).3C2, (C2×C8.C4)⋊4C2, C4○D4.27(C2×C4), (C2×C4).231(C2×D4), C4.93(C2×C22⋊C4), (C2×C4).739(C4○D4), (C2×C4).178(C22×C4), (C2×C4○D4).258C22, SmallGroup(128,546)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8○D4⋊C4
G = < a,b,c,d | a8=c2=d4=1, b2=a4, ab=ba, ac=ca, dad-1=a-1, cbc=a4b, dbd-1=a6c, dcd-1=a2b >
Subgroups: 228 in 128 conjugacy classes, 58 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4≀C2, C4.Q8, C2.D8, C8.C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, M4(2)⋊4C4, C42⋊C22, C23.25D4, C2×C8.C4, C2×C8○D4, C8○D4⋊C4
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, C8○D4⋊C4
►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)
(1 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 14)(10 13)(11 12)(15 16)(17 18)(19 24)(20 23)(21 22)
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), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,26,5,30)(2,25,6,29)(3,32,7,28)(4,31,8,27)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22)>;
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), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,26,5,30)(2,25,6,29)(3,32,7,28)(4,31,8,27)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,23)(21,22) );
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)], [(1,12,5,16),(2,13,6,9),(3,14,7,10),(4,15,8,11),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25)], [(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25)], [(1,26,5,30),(2,25,6,29),(3,32,7,28),(4,31,8,27),(9,14),(10,13),(11,12),(15,16),(17,18),(19,24),(20,23),(21,22)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D4 | C4○D4 | C4○D4 | C8○D4⋊C4 |
| kernel | C8○D4⋊C4 | M4(2)⋊4C4 | C42⋊C22 | C23.25D4 | C2×C8.C4 | C2×C8○D4 | C8○D4 | C2×C8 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C8○D4⋊C4 ►in GL4(𝔽17) generated by
| 11 | 11 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(17))| [11,3,0,0,11,0,0,0,0,0,11,3,0,0,11,0],[0,0,13,0,0,0,0,13,13,0,0,0,0,13,0,0],[0,0,16,1,0,0,15,1,1,16,0,0,2,16,0,0],[0,5,0,0,10,0,0,0,0,0,0,3,0,0,6,0] >;
C8○D4⋊C4 in GAP, Magma, Sage, TeX
C_8\circ D_4\rtimes C_4
% in TeX
G:=Group("C8oD4:C4"); // GroupNames label
G:=SmallGroup(128,546);
// by ID
G=gap.SmallGroup(128,546);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,718,172,1027]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^2=d^4=1,b^2=a^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^4*b,d*b*d^-1=a^6*c,d*c*d^-1=a^2*b>;
// generators/relations