direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C16⋊C22, D16⋊2C6, C48⋊6C22, SD32⋊1C6, C24.48D4, C12.65D8, M5(2)⋊1C6, C24.67C23, C16⋊(C2×C6), C4○D8⋊2C6, D8⋊2(C2×C6), C8.3(C3×D4), (C6×D8)⋊24C2, (C3×D16)⋊6C2, (C2×D8)⋊10C6, Q16⋊2(C2×C6), C4.14(C3×D8), C2.16(C6×D8), C4.11(C6×D4), C6.88(C2×D8), (C2×C6).27D8, (C3×SD32)⋊5C2, C8.7(C22×C6), C22.5(C3×D8), C12.318(C2×D4), (C2×C12).346D4, (C3×D8)⋊18C22, (C3×M5(2))⋊3C2, (C3×Q16)⋊16C22, (C2×C24).206C22, (C3×C4○D8)⋊9C2, (C2×C8).30(C2×C6), (C2×C4).47(C3×D4), SmallGroup(192,942)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C16⋊C22
G = < a,b,c,d | a3=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >
Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, M5(2), D16, SD32, C2×D8, C4○D8, C48, C2×C24, C3×D8, C3×D8, C3×D8, C3×SD16, C3×Q16, C6×D4, C3×C4○D4, C16⋊C22, C3×M5(2), C3×D16, C3×SD32, C6×D8, C3×C4○D8, C3×C16⋊C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, C16⋊C22, C6×D8, C3×C16⋊C22
►On 48 points
(1 19 43)(2 20 44)(3 21 45)(4 22 46)(5 23 47)(6 24 48)(7 25 33)(8 26 34)(9 27 35)(10 28 36)(11 29 37)(12 30 38)(13 31 39)(14 32 40)(15 17 41)(16 18 42)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 37)(34 44)(36 42)(38 40)(39 47)(41 45)(46 48)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)
G:=sub<Sym(48)| (1,19,43)(2,20,44)(3,21,45)(4,22,46)(5,23,47)(6,24,48)(7,25,33)(8,26,34)(9,27,35)(10,28,36)(11,29,37)(12,30,38)(13,31,39)(14,32,40)(15,17,41)(16,18,42), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47)>;
G:=Group( (1,19,43)(2,20,44)(3,21,45)(4,22,46)(5,23,47)(6,24,48)(7,25,33)(8,26,34)(9,27,35)(10,28,36)(11,29,37)(12,30,38)(13,31,39)(14,32,40)(15,17,41)(16,18,42), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,21)(18,28)(20,26)(22,24)(23,31)(25,29)(30,32)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47) );
G=PermutationGroup([[(1,19,43),(2,20,44),(3,21,45),(4,22,46),(5,23,47),(6,24,48),(7,25,33),(8,26,34),(9,27,35),(10,28,36),(11,29,37),(12,30,38),(13,31,39),(14,32,40),(15,17,41),(16,18,42)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,21),(18,28),(20,26),(22,24),(23,31),(25,29),(30,32),(33,37),(34,44),(36,42),(38,40),(39,47),(41,45),(46,48)], [(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31),(33,41),(35,43),(37,45),(39,47)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | 12F | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D8 | D8 | C3×D4 | C3×D4 | C3×D8 | C3×D8 | C16⋊C22 | C3×C16⋊C22 |
| kernel | C3×C16⋊C22 | C3×M5(2) | C3×D16 | C3×SD32 | C6×D8 | C3×C4○D8 | C16⋊C22 | M5(2) | D16 | SD32 | C2×D8 | C4○D8 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C16⋊C22 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 6 | 4 | 2 |
| 4 | 6 | 5 | 1 |
| 6 | 0 | 5 | 0 |
| 6 | 1 | 3 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 6 | 3 |
| 0 | 2 | 3 | 2 |
| 0 | 3 | 2 | 2 |
| 0 | 1 | 1 | 2 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,4,6,6,6,6,0,1,4,5,5,3,2,1,0,2],[1,0,0,0,0,6,0,0,0,0,0,2,0,0,4,0],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;
C3×C16⋊C22 in GAP, Magma, Sage, TeX
C_3\times C_{16}\rtimes C_2^2 % in TeX
G:=Group("C3xC16:C2^2"); // GroupNames label
G:=SmallGroup(192,942);
// by ID
G=gap.SmallGroup(192,942);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,2102,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations