direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C22⋊D8, (C2×C6)⋊7D8, (C2×D8)⋊1C6, D4⋊2(C3×D4), C2.4(C6×D8), (C3×D4)⋊20D4, (C6×D8)⋊15C2, C4⋊D4⋊1C6, C22⋊C8⋊3C6, C6.76(C2×D8), C4.21(C6×D4), C22⋊3(C3×D8), D4⋊C4⋊4C6, (C22×D4)⋊5C6, (C2×C24)⋊20C22, C6.94C22≀C2, (C2×C12).317D4, C12.382(C2×D4), (C6×D4)⋊27C22, C23.46(C3×D4), C22.77(C6×D4), (C22×C6).163D4, C6.131(C8⋊C22), (C2×C12).912C23, (C22×C12).419C22, C4⋊C4⋊1(C2×C6), (C2×C8)⋊1(C2×C6), (D4×C2×C6)⋊14C2, (C2×D4)⋊1(C2×C6), (C2×C4).26(C3×D4), C2.6(C3×C8⋊C22), (C3×C4⋊D4)⋊28C2, (C3×C4⋊C4)⋊35C22, (C3×C22⋊C8)⋊13C2, C2.8(C3×C22≀C2), (C2×C6).633(C2×D4), (C3×D4⋊C4)⋊15C2, (C2×C4).87(C22×C6), (C22×C4).42(C2×C6), SmallGroup(192,880)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22⋊D8
G = < a,b,c,d,e | a3=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 450 in 198 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C22×C12, C6×D4, C6×D4, C6×D4, C23×C6, C22⋊D8, C3×C22⋊C8, C3×D4⋊C4, C3×C4⋊D4, C6×D8, D4×C2×C6, C3×C22⋊D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C22≀C2, C2×D8, C8⋊C22, C3×D8, C6×D4, C22⋊D8, C3×C22≀C2, C6×D8, C3×C8⋊C22, C3×C22⋊D8
►On 48 points
(1 13 39)(2 14 40)(3 15 33)(4 16 34)(5 9 35)(6 10 36)(7 11 37)(8 12 38)(17 32 42)(18 25 43)(19 26 44)(20 27 45)(21 28 46)(22 29 47)(23 30 48)(24 31 41)
(1 22)(3 24)(5 18)(7 20)(9 25)(11 27)(13 29)(15 31)(33 41)(35 43)(37 45)(39 47)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)
G:=sub<Sym(48)| (1,13,39)(2,14,40)(3,15,33)(4,16,34)(5,9,35)(6,10,36)(7,11,37)(8,12,38)(17,32,42)(18,25,43)(19,26,44)(20,27,45)(21,28,46)(22,29,47)(23,30,48)(24,31,41), (1,22)(3,24)(5,18)(7,20)(9,25)(11,27)(13,29)(15,31)(33,41)(35,43)(37,45)(39,47), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)>;
G:=Group( (1,13,39)(2,14,40)(3,15,33)(4,16,34)(5,9,35)(6,10,36)(7,11,37)(8,12,38)(17,32,42)(18,25,43)(19,26,44)(20,27,45)(21,28,46)(22,29,47)(23,30,48)(24,31,41), (1,22)(3,24)(5,18)(7,20)(9,25)(11,27)(13,29)(15,31)(33,41)(35,43)(37,45)(39,47), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47) );
G=PermutationGroup([[(1,13,39),(2,14,40),(3,15,33),(4,16,34),(5,9,35),(6,10,36),(7,11,37),(8,12,38),(17,32,42),(18,25,43),(19,26,44),(20,27,45),(21,28,46),(22,29,47),(23,30,48),(24,31,41)], [(1,22),(3,24),(5,18),(7,20),(9,25),(11,27),(13,29),(15,31),(33,41),(35,43),(37,45),(39,47)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,18),(19,24),(20,23),(21,22),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,40),(38,39),(41,44),(42,43),(45,48),(46,47)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 6S | 6T | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
57 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 | D4 | D8 | C3×D4 | C3×D4 | C3×D4 | C3×D8 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C22⋊D8 | C3×C22⋊C8 | C3×D4⋊C4 | C3×C4⋊D4 | C6×D8 | D4×C2×C6 | C22⋊D8 | C22⋊C8 | D4⋊C4 | C4⋊D4 | C2×D8 | C22×D4 | C2×C12 | C3×D4 | C22×C6 | C2×C6 | C2×C4 | D4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 8 | 1 | 2 |
Matrix representation of C3×C22⋊D8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 41 | 0 | 0 |
| 16 | 41 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 1 |
| 0 | 41 | 0 | 0 |
| 57 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,72,1,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,16,0,0,41,41,0,0,0,0,72,0,0,0,71,1],[0,57,0,0,41,0,0,0,0,0,72,0,0,0,71,1] >;
C3×C22⋊D8 in GAP, Magma, Sage, TeX
C_3\times C_2^2\rtimes D_8
% in TeX
G:=Group("C3xC2^2:D8"); // GroupNames label
G:=SmallGroup(192,880);
// by ID
G=gap.SmallGroup(192,880);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations