direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×M5(2)⋊C2, D8.2C12, C12.62D8, C24.90D4, M5(2)⋊6C6, C8.2(C2×C12), (C2×D8).5C6, (C3×D8).4C4, C8.16(C3×D4), C4.11(C3×D8), C8.C4⋊2C6, C24.37(C2×C4), (C6×D8).12C2, (C2×C12).280D4, (C2×C6).24SD16, (C3×M5(2))⋊14C2, C6.41(D4⋊C4), C22.3(C3×SD16), C12.73(C22⋊C4), (C2×C24).194C22, (C2×C8).13(C2×C6), (C2×C4).11(C3×D4), C4.5(C3×C22⋊C4), (C3×C8.C4)⋊11C2, C2.10(C3×D4⋊C4), SmallGroup(192,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M5(2)⋊C2
G = < a,b,c,d | a3=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >
Subgroups: 162 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C8, C2×C4, D4, C23, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), D8, D8, C2×D4, C24, C24, C2×C12, C3×D4, C22×C6, C8.C4, M5(2), C2×D8, C48, C2×C24, C3×M4(2), C3×D8, C3×D8, C6×D4, M5(2)⋊C2, C3×C8.C4, C3×M5(2), C6×D8, C3×M5(2)⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, D4⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, M5(2)⋊C2, C3×D4⋊C4, C3×M5(2)⋊C2
►On 48 points
(1 42 17)(2 43 18)(3 44 19)(4 45 20)(5 46 21)(6 47 22)(7 48 23)(8 33 24)(9 34 25)(10 35 26)(11 36 27)(12 37 28)(13 38 29)(14 39 30)(15 40 31)(16 41 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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)
(2 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(18 28)(19 31)(20 26)(21 29)(22 24)(23 27)(30 32)(33 47)(35 45)(36 48)(37 43)(38 46)(39 41)(40 44)
G:=sub<Sym(48)| (1,42,17)(2,43,18)(3,44,19)(4,45,20)(5,46,21)(6,47,22)(7,48,23)(8,33,24)(9,34,25)(10,35,26)(11,36,27)(12,37,28)(13,38,29)(14,39,30)(15,40,31)(16,41,32), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(18,28)(19,31)(20,26)(21,29)(22,24)(23,27)(30,32)(33,47)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44)>;
G:=Group( (1,42,17)(2,43,18)(3,44,19)(4,45,20)(5,46,21)(6,47,22)(7,48,23)(8,33,24)(9,34,25)(10,35,26)(11,36,27)(12,37,28)(13,38,29)(14,39,30)(15,40,31)(16,41,32), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(18,28)(19,31)(20,26)(21,29)(22,24)(23,27)(30,32)(33,47)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44) );
G=PermutationGroup([[(1,42,17),(2,43,18),(3,44,19),(4,45,20),(5,46,21),(6,47,22),(7,48,23),(8,33,24),(9,34,25),(10,35,26),(11,36,27),(12,37,28),(13,38,29),(14,39,30),(15,40,31),(16,41,32)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47)], [(2,12),(3,15),(4,10),(5,13),(6,8),(7,11),(14,16),(18,28),(19,31),(20,26),(21,29),(22,24),(23,27),(30,32),(33,47),(35,45),(36,48),(37,43),(38,46),(39,41),(40,44)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 24I | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 | M5(2)⋊C2 | C3×M5(2)⋊C2 |
| kernel | C3×M5(2)⋊C2 | C3×C8.C4 | C3×M5(2) | C6×D8 | M5(2)⋊C2 | C3×D8 | C8.C4 | M5(2) | C2×D8 | D8 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×M5(2)⋊C2 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 1 | 6 | 1 |
| 1 | 2 | 1 | 1 |
| 0 | 2 | 5 | 3 |
| 2 | 5 | 5 | 3 |
| 1 | 0 | 6 | 3 |
| 0 | 2 | 3 | 2 |
| 0 | 3 | 2 | 2 |
| 0 | 1 | 1 | 2 |
| 1 | 0 | 6 | 3 |
| 0 | 5 | 4 | 5 |
| 0 | 4 | 4 | 1 |
| 0 | 6 | 4 | 4 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,0,2,1,2,2,5,6,1,5,5,1,1,3,3],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2],[1,0,0,0,0,5,4,6,6,4,4,4,3,5,1,4] >;
C3×M5(2)⋊C2 in GAP, Magma, Sage, TeX
C_3\times M_5(2)\rtimes C_2
% in TeX
G:=Group("C3xM5(2):C2"); // GroupNames label
G:=SmallGroup(192,167);
// by ID
G=gap.SmallGroup(192,167);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,2194,136,2111,172,6053,3036,124]);
// 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^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations