p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C4⋊C16, C8.7Q8, C8.28D4, C42.7C4, C2.3M5(2), C4.11M4(2), (C4×C8).2C2, (C2×C4).4C8, (C2×C8).7C4, C2.2(C4⋊C8), (C2×C16).2C2, C2.2(C2×C16), C4.19(C4⋊C4), C22.9(C2×C8), (C2×C8).108C22, (C2×C4).82(C2×C4), SmallGroup(64,44)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C16
G = < a,b | a4=b16=1, bab-1=a-1 >
Smallest permutation representation of C4⋊C16
►Regular action on 64 points
(1 32 38 58)(2 59 39 17)(3 18 40 60)(4 61 41 19)(5 20 42 62)(6 63 43 21)(7 22 44 64)(8 49 45 23)(9 24 46 50)(10 51 47 25)(11 26 48 52)(12 53 33 27)(13 28 34 54)(14 55 35 29)(15 30 36 56)(16 57 37 31)
(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)
G:=sub<Sym(64)| (1,32,38,58)(2,59,39,17)(3,18,40,60)(4,61,41,19)(5,20,42,62)(6,63,43,21)(7,22,44,64)(8,49,45,23)(9,24,46,50)(10,51,47,25)(11,26,48,52)(12,53,33,27)(13,28,34,54)(14,55,35,29)(15,30,36,56)(16,57,37,31), (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)>;
G:=Group( (1,32,38,58)(2,59,39,17)(3,18,40,60)(4,61,41,19)(5,20,42,62)(6,63,43,21)(7,22,44,64)(8,49,45,23)(9,24,46,50)(10,51,47,25)(11,26,48,52)(12,53,33,27)(13,28,34,54)(14,55,35,29)(15,30,36,56)(16,57,37,31), (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) );
G=PermutationGroup([[(1,32,38,58),(2,59,39,17),(3,18,40,60),(4,61,41,19),(5,20,42,62),(6,63,43,21),(7,22,44,64),(8,49,45,23),(9,24,46,50),(10,51,47,25),(11,26,48,52),(12,53,33,27),(13,28,34,54),(14,55,35,29),(15,30,36,56),(16,57,37,31)], [(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)]])
C4⋊C16 is a maximal subgroup of
C8.31D8 C8.17Q16 C8.30D8 C4.D16 C8.27D8 C8.16Q16 C4.10D16 C4.6Q32 C4⋊M5(2) C4⋊C4.7C8 C8.12M4(2) D4×C16 C16⋊9D4 C16⋊6D4 Q8×C16 D8⋊2D4 Q16⋊2D4 D8.4D4 Q16.4D4 D8.5D4 Q16.5D4 D8⋊1Q8 Q16⋊Q8 D8⋊Q8 C4.Q32 D8.Q8 Q16.Q8
C4p⋊C16: C8⋊2C16 C8.36D8 C12⋊C16 C20⋊3C16 C20⋊C16 C28⋊C16 ...
C2p.M5(2): D4⋊C16 Q8⋊C16 C42.13C8 C42.6C8 C16⋊4Q8 Dic3⋊C16 C40.88D4 C10.M5(2) ...
C4⋊C16 is a maximal quotient of
C22.7M5(2) C10.M5(2)
C4p⋊C16: C8⋊2C16 C8.36D8 C12⋊C16 C20⋊3C16 C20⋊C16 C28⋊C16 ...
C8p.D4: C4⋊C32 C8.C16 Dic3⋊C16 C40.88D4 Dic7⋊C16 ...
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | M4(2) | M5(2) |
| kernel | C4⋊C16 | C4×C8 | C2×C16 | C42 | C2×C8 | C2×C4 | C4 | C8 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C4⋊C16 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 1 | 13 |
| 0 | 9 | 16 |
| 14 | 0 | 0 |
| 0 | 10 | 11 |
| 0 | 0 | 7 |
G:=sub<GL(3,GF(17))| [1,0,0,0,1,9,0,13,16],[14,0,0,0,10,0,0,11,7] >;
C4⋊C16 in GAP, Magma, Sage, TeX
C_4\rtimes C_{16} % in TeX
G:=Group("C4:C16"); // GroupNames label
G:=SmallGroup(64,44);
// by ID
G=gap.SmallGroup(64,44);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,69,88]);
// Polycyclic
G:=Group<a,b|a^4=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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