p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C16, C8.27D4, C23.2C8, C2.2M5(2), C4.10M4(2), (C2×C16)⋊1C2, (C2×C4).3C8, (C2×C8).6C4, C2.1(C2×C16), (C22×C8).4C2, (C22×C4).8C4, C22.8(C2×C8), C2.2(C22⋊C8), C4.28(C22⋊C4), (C2×C8).107C22, (C2×C4).81(C2×C4), SmallGroup(64,29)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22⋊C16
G = < a,b,c | a2=b2=c16=1, cac-1=ab=ba, bc=cb >
Smallest permutation representation of C22⋊C16
►On 32 points
(1 9)(2 20)(3 11)(4 22)(5 13)(6 24)(7 15)(8 26)(10 28)(12 30)(14 32)(16 18)(17 25)(19 27)(21 29)(23 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)
(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)
G:=sub<Sym(32)| (1,9)(2,20)(3,11)(4,22)(5,13)(6,24)(7,15)(8,26)(10,28)(12,30)(14,32)(16,18)(17,25)(19,27)(21,29)(23,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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)>;
G:=Group( (1,9)(2,20)(3,11)(4,22)(5,13)(6,24)(7,15)(8,26)(10,28)(12,30)(14,32)(16,18)(17,25)(19,27)(21,29)(23,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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) );
G=PermutationGroup([[(1,9),(2,20),(3,11),(4,22),(5,13),(6,24),(7,15),(8,26),(10,28),(12,30),(14,32),(16,18),(17,25),(19,27),(21,29),(23,31)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26)], [(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)]])
C22⋊C16 is a maximal subgroup of
C23⋊C16 C23.M4(2) C22.M5(2) C23.7M4(2) C22.SD32 C23.32D8 C23.12SD16 C23.13SD16 C24.5C8 (C2×D4).5C8 C8.12M4(2) D4×C16 C16⋊9D4 C16⋊6D4 D8⋊7D4 Q16⋊7D4 D8⋊8D4 D8.9D4 Q16.8D4 D8.10D4 C22.D16 C23.49D8 C23.19D8 C23.50D8 C23.51D8 C23.20D8 A4⋊C16
C2p.M5(2): C42.13C8 C42.6C8 D6⋊C16 C24.98D4 D10⋊1C16 C40.91D4 D10⋊C16 C10.6M5(2) ...
C22⋊C16 is a maximal quotient of
C23⋊C16 C22.M5(2) Q8⋊C16 C22.7M5(2) C10.6M5(2)
D2p⋊C16: D4⋊C16 D6⋊C16 D10⋊1C16 D10⋊C16 D14⋊C16 ...
C8p.D4: C22⋊C32 C23.C16 D4.C16 C24.98D4 C40.91D4 C56.91D4 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | M4(2) | M5(2) |
| kernel | C22⋊C16 | C2×C16 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C22⋊C16 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 4 | 16 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 3 | 0 | 0 |
| 0 | 4 | 15 |
| 0 | 0 | 13 |
G:=sub<GL(3,GF(17))| [1,0,0,0,1,4,0,0,16],[1,0,0,0,16,0,0,0,16],[3,0,0,0,4,0,0,15,13] >;
C22⋊C16 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_{16} % in TeX
G:=Group("C2^2:C16"); // GroupNames label
G:=SmallGroup(64,29);
// by ID
G=gap.SmallGroup(64,29);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,69,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^16=1,c*a*c^-1=a*b=b*a,b*c=c*b>;
// generators/relations
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