p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42⋊16Q8, C43.10C2, C42.328D4, C4⋊Q8⋊27C4, C4.17C4≀C2, C4.16(C4×Q8), C4.54(C4⋊Q8), C42.C2⋊11C4, C42.271(C2×C4), C23.574(C2×D4), (C22×C4).766D4, C42⋊6C4.11C2, C4.94(C4.4D4), C4⋊M4(2).33C2, C22.28(C22⋊Q8), C42⋊C2.44C22, (C2×C42).1081C22, (C22×C4).1428C23, (C2×M4(2)).216C22, C23.37C23.20C2, C2.16(C23.67C23), C2.46(C2×C4≀C2), C4⋊C4.101(C2×C4), (C2×C4).215(C2×Q8), (C2×C4).1549(C2×D4), (C2×C4).611(C4○D4), (C2×C4).442(C22×C4), (C2×C4).208(C22⋊C4), C22.303(C2×C22⋊C4), SmallGroup(128,726)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊16Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 244 in 142 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C4⋊C8, C2×C42, C2×C42, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C42⋊6C4, C43, C4⋊M4(2), C23.37C23, C42⋊16Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4≀C2, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C23.67C23, C2×C4≀C2, C42⋊16Q8
►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 27 18 23)(2 28 19 24)(3 25 20 21)(4 26 17 22)(5 11 30 13)(6 12 31 14)(7 9 32 15)(8 10 29 16)
(1 19 3 17)(2 20 4 18)(5 14 32 10)(6 15 29 11)(7 16 30 12)(8 13 31 9)(21 26 23 28)(22 27 24 25)
(1 29 3 6)(2 9 4 13)(5 28 32 26)(7 22 30 24)(8 20 31 18)(10 21 14 23)(11 19 15 17)(12 27 16 25)
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), (1,27,18,23)(2,28,19,24)(3,25,20,21)(4,26,17,22)(5,11,30,13)(6,12,31,14)(7,9,32,15)(8,10,29,16), (1,19,3,17)(2,20,4,18)(5,14,32,10)(6,15,29,11)(7,16,30,12)(8,13,31,9)(21,26,23,28)(22,27,24,25), (1,29,3,6)(2,9,4,13)(5,28,32,26)(7,22,30,24)(8,20,31,18)(10,21,14,23)(11,19,15,17)(12,27,16,25)>;
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), (1,27,18,23)(2,28,19,24)(3,25,20,21)(4,26,17,22)(5,11,30,13)(6,12,31,14)(7,9,32,15)(8,10,29,16), (1,19,3,17)(2,20,4,18)(5,14,32,10)(6,15,29,11)(7,16,30,12)(8,13,31,9)(21,26,23,28)(22,27,24,25), (1,29,3,6)(2,9,4,13)(5,28,32,26)(7,22,30,24)(8,20,31,18)(10,21,14,23)(11,19,15,17)(12,27,16,25) );
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)], [(1,27,18,23),(2,28,19,24),(3,25,20,21),(4,26,17,22),(5,11,30,13),(6,12,31,14),(7,9,32,15),(8,10,29,16)], [(1,19,3,17),(2,20,4,18),(5,14,32,10),(6,15,29,11),(7,16,30,12),(8,13,31,9),(21,26,23,28),(22,27,24,25)], [(1,29,3,6),(2,9,4,13),(5,28,32,26),(7,22,30,24),(8,20,31,18),(10,21,14,23),(11,19,15,17),(12,27,16,25)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4AD | 4AE | 4AF | 4AG | 4AH | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | C4○D4 | C4≀C2 |
| kernel | C42⋊16Q8 | C42⋊6C4 | C43 | C4⋊M4(2) | C23.37C23 | C42.C2 | C4⋊Q8 | C42 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 4 | 16 |
Matrix representation of C42⋊16Q8 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 5 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,4,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,13,0,0,0,0,4],[5,5,0,0,5,12,0,0,0,0,0,4,0,0,4,0] >;
C42⋊16Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{16}Q_8 % in TeX
G:=Group("C4^2:16Q8"); // GroupNames label
G:=SmallGroup(128,726);
// by ID
G=gap.SmallGroup(128,726);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,248,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations