direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×S3×D4, C42⋊34D6, C4⋊C4⋊55D6, (D4×C12)⋊8C2, C12⋊13(C2×D4), (C4×D12)⋊23C2, D12⋊12(C2×C4), (S3×C42)⋊3C2, C12⋊1(C22×C4), C22⋊C4⋊52D6, D6.58(C2×D4), D6⋊4(C22×C4), (C22×C4)⋊39D6, (C4×C12)⋊15C22, D6⋊C4⋊61C22, (D4×Dic3)⋊44C2, Dic3⋊11(C2×D4), (C2×D4).244D6, C6.21(C23×C4), (C2×C6).88C24, Dic3⋊5D4⋊46C2, C6.46(C22×D4), D6.35(C4○D4), C4⋊Dic3⋊72C22, Dic3⋊2(C22×C4), Dic3⋊4D4⋊51C2, (C2×C12).586C23, Dic3⋊C4⋊63C22, (C22×C12)⋊35C22, (C4×Dic3)⋊78C22, (C6×D4).252C22, C22.31(S3×C23), (C2×D12).257C22, C6.D4⋊47C22, C23.177(C22×S3), (C22×C6).158C23, (S3×C23).105C22, (C22×S3).254C23, (C2×Dic3).308C23, (C22×Dic3)⋊43C22, C3⋊4(C2×C4×D4), C4⋊1(S3×C2×C4), C2.5(C2×S3×D4), C22⋊2(S3×C2×C4), (S3×C4⋊C4)⋊47C2, (C4×S3)⋊7(C2×C4), C3⋊D4⋊2(C2×C4), (C2×S3×D4).11C2, C2.4(S3×C4○D4), (C3×D4)⋊11(C2×C4), (C4×C3⋊D4)⋊39C2, (S3×C2×C4)⋊69C22, (S3×C22×C4)⋊21C2, (C2×C6)⋊1(C22×C4), C2.23(S3×C22×C4), (C3×C4⋊C4)⋊55C22, (S3×C22⋊C4)⋊30C2, C6.138(C2×C4○D4), (C22×S3)⋊12(C2×C4), (C3×C22⋊C4)⋊62C22, (C2×C4).819(C22×S3), (C2×C3⋊D4).109C22, SmallGroup(192,1103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×S3×D4
G = < a,b,c,d,e | a4=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1032 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C2×C4×D4, S3×C42, C4×D12, S3×C22⋊C4, Dic3⋊4D4, S3×C4⋊C4, Dic3⋊5D4, C4×C3⋊D4, D4×Dic3, D4×C12, S3×C22×C4, C2×S3×D4, C4×S3×D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C24, C4×S3, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×C2×C4, S3×D4, S3×C23, C2×C4×D4, S3×C22×C4, C2×S3×D4, S3×C4○D4, C4×S3×D4
►On 48 points
(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 25 22)(2 26 23)(3 27 24)(4 28 21)(5 29 19)(6 30 20)(7 31 17)(8 32 18)(9 43 37)(10 44 38)(11 41 39)(12 42 40)(13 34 48)(14 35 45)(15 36 46)(16 33 47)
(1 3)(2 4)(5 7)(6 8)(9 41)(10 42)(11 43)(12 44)(13 36)(14 33)(15 34)(16 35)(17 29)(18 30)(19 31)(20 32)(21 26)(22 27)(23 28)(24 25)(37 39)(38 40)(45 47)(46 48)
(1 45 39 6)(2 46 40 7)(3 47 37 8)(4 48 38 5)(9 32 27 16)(10 29 28 13)(11 30 25 14)(12 31 26 15)(17 23 36 42)(18 24 33 43)(19 21 34 44)(20 22 35 41)
(5 48)(6 45)(7 46)(8 47)(13 29)(14 30)(15 31)(16 32)(17 36)(18 33)(19 34)(20 35)
G:=sub<Sym(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), (1,25,22)(2,26,23)(3,27,24)(4,28,21)(5,29,19)(6,30,20)(7,31,17)(8,32,18)(9,43,37)(10,44,38)(11,41,39)(12,42,40)(13,34,48)(14,35,45)(15,36,46)(16,33,47), (1,3)(2,4)(5,7)(6,8)(9,41)(10,42)(11,43)(12,44)(13,36)(14,33)(15,34)(16,35)(17,29)(18,30)(19,31)(20,32)(21,26)(22,27)(23,28)(24,25)(37,39)(38,40)(45,47)(46,48), (1,45,39,6)(2,46,40,7)(3,47,37,8)(4,48,38,5)(9,32,27,16)(10,29,28,13)(11,30,25,14)(12,31,26,15)(17,23,36,42)(18,24,33,43)(19,21,34,44)(20,22,35,41), (5,48)(6,45)(7,46)(8,47)(13,29)(14,30)(15,31)(16,32)(17,36)(18,33)(19,34)(20,35)>;
G:=Group( (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), (1,25,22)(2,26,23)(3,27,24)(4,28,21)(5,29,19)(6,30,20)(7,31,17)(8,32,18)(9,43,37)(10,44,38)(11,41,39)(12,42,40)(13,34,48)(14,35,45)(15,36,46)(16,33,47), (1,3)(2,4)(5,7)(6,8)(9,41)(10,42)(11,43)(12,44)(13,36)(14,33)(15,34)(16,35)(17,29)(18,30)(19,31)(20,32)(21,26)(22,27)(23,28)(24,25)(37,39)(38,40)(45,47)(46,48), (1,45,39,6)(2,46,40,7)(3,47,37,8)(4,48,38,5)(9,32,27,16)(10,29,28,13)(11,30,25,14)(12,31,26,15)(17,23,36,42)(18,24,33,43)(19,21,34,44)(20,22,35,41), (5,48)(6,45)(7,46)(8,47)(13,29)(14,30)(15,31)(16,32)(17,36)(18,33)(19,34)(20,35) );
G=PermutationGroup([[(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)], [(1,25,22),(2,26,23),(3,27,24),(4,28,21),(5,29,19),(6,30,20),(7,31,17),(8,32,18),(9,43,37),(10,44,38),(11,41,39),(12,42,40),(13,34,48),(14,35,45),(15,36,46),(16,33,47)], [(1,3),(2,4),(5,7),(6,8),(9,41),(10,42),(11,43),(12,44),(13,36),(14,33),(15,34),(16,35),(17,29),(18,30),(19,31),(20,32),(21,26),(22,27),(23,28),(24,25),(37,39),(38,40),(45,47),(46,48)], [(1,45,39,6),(2,46,40,7),(3,47,37,8),(4,48,38,5),(9,32,27,16),(10,29,28,13),(11,30,25,14),(12,31,26,15),(17,23,36,42),(18,24,33,43),(19,21,34,44),(20,22,35,41)], [(5,48),(6,45),(7,46),(8,47),(13,29),(14,30),(15,31),(16,32),(17,36),(18,33),(19,34),(20,35)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 4Q | ··· | 4X | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | S3×C4○D4 |
| kernel | C4×S3×D4 | S3×C42 | C4×D12 | S3×C22⋊C4 | Dic3⋊4D4 | S3×C4⋊C4 | Dic3⋊5D4 | C4×C3⋊D4 | D4×Dic3 | D4×C12 | S3×C22×C4 | C2×S3×D4 | S3×D4 | C4×D4 | C4×S3 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 16 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C4×S3×D4 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,12,1,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
C4×S3×D4 in GAP, Magma, Sage, TeX
C_4\times S_3\times D_4
% in TeX
G:=Group("C4xS3xD4"); // GroupNames label
G:=SmallGroup(192,1103);
// by ID
G=gap.SmallGroup(192,1103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations