p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C16⋊5C4, C8.2C8, C42.4C4, C4.6C42, C2.1M5(2), C2.3(C4×C8), (C2×C4).2C8, C4.12(C2×C8), C8.22(C2×C4), (C2×C8).12C4, (C2×C16).7C2, (C4×C8).14C2, C22.7(C2×C8), (C2×C8).106C22, (C2×C4).80(C2×C4), SmallGroup(64,27)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊5C4
G = < a,b | a16=b4=1, bab-1=a9 >
Smallest permutation representation of C16⋊5C4
►Regular action on 64 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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 61 45 25)(2 54 46 18)(3 63 47 27)(4 56 48 20)(5 49 33 29)(6 58 34 22)(7 51 35 31)(8 60 36 24)(9 53 37 17)(10 62 38 26)(11 55 39 19)(12 64 40 28)(13 57 41 21)(14 50 42 30)(15 59 43 23)(16 52 44 32)
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,61,45,25)(2,54,46,18)(3,63,47,27)(4,56,48,20)(5,49,33,29)(6,58,34,22)(7,51,35,31)(8,60,36,24)(9,53,37,17)(10,62,38,26)(11,55,39,19)(12,64,40,28)(13,57,41,21)(14,50,42,30)(15,59,43,23)(16,52,44,32)>;
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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,61,45,25)(2,54,46,18)(3,63,47,27)(4,56,48,20)(5,49,33,29)(6,58,34,22)(7,51,35,31)(8,60,36,24)(9,53,37,17)(10,62,38,26)(11,55,39,19)(12,64,40,28)(13,57,41,21)(14,50,42,30)(15,59,43,23)(16,52,44,32) );
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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,61,45,25),(2,54,46,18),(3,63,47,27),(4,56,48,20),(5,49,33,29),(6,58,34,22),(7,51,35,31),(8,60,36,24),(9,53,37,17),(10,62,38,26),(11,55,39,19),(12,64,40,28),(13,57,41,21),(14,50,42,30),(15,59,43,23),(16,52,44,32)]])
C16⋊5C4 is a maximal subgroup of
C16⋊C8 C8.31D8 D8⋊C8 Q16⋊C8 C8.17Q16 C16⋊1C8 C4×M5(2) C16○2M5(2) C42.6C8 C8.12M4(2) C16⋊9D4 D8.C8 SD32⋊3C4 Q32⋊4C4 D16⋊4C4 D16⋊5C4 C16⋊4Q8 C8.12SD16 C8.13SD16 C8.14SD16 C16⋊3D4 C8.7D8 C16⋊Q8 C42.4F5
C16p⋊C4: C32⋊C4 C48⋊10C4 C80⋊17C4 C16⋊7F5 C112⋊9C4 ...
C8p.C8: C16.C8 C24.C8 C40.10C8 C40.C8 C56.C8 ...
C16⋊5C4 is a maximal quotient of
C22.7M5(2)
C16p⋊C4: C32⋊C4 C48⋊10C4 C80⋊17C4 C16⋊7F5 C112⋊9C4 ...
C2p.M5(2): C16⋊5C8 C8⋊C16 C24.C8 C40.10C8 C42.4F5 C40.C8 C56.C8 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | M5(2) |
| kernel | C16⋊5C4 | C4×C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C16⋊5C4 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 16 | 4 |
| 0 | 13 | 1 |
| 13 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(17))| [13,0,0,0,16,13,0,4,1],[13,0,0,0,0,1,0,1,0] >;
C16⋊5C4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_5C_4 % in TeX
G:=Group("C16:5C4"); // GroupNames label
G:=SmallGroup(64,27);
// by ID
G=gap.SmallGroup(64,27);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,24,409,55,86,88]);
// Polycyclic
G:=Group<a,b|a^16=b^4=1,b*a*b^-1=a^9>;
// generators/relations
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