direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×S3×D4, C42⋊34D6, C4⋊C4⋊55D6, (D4×C12)⋊8C2, C12⋊13(C2×D4), D12⋊12(C2×C4), (C4×D12)⋊23C2, (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, Dic3⋊11(C2×D4), (D4×Dic3)⋊44C2, (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, (C22×C6).158C23, C23.177(C22×S3), (C22×S3).254C23, (S3×C23).105C22, (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
Subgroups: 1032 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×4], C4 [×10], C22, C22 [×4], C22 [×34], S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×35], D4 [×4], D4 [×12], C23 [×2], C23 [×19], Dic3 [×4], Dic3 [×3], C12 [×4], C12 [×3], D6 [×10], D6 [×20], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×19], C2×D4, C2×D4 [×11], C24 [×2], C4×S3 [×8], C4×S3 [×14], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×8], C22×C6 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4, C4×D4 [×7], C23×C4 [×2], C22×D4, C4×Dic3 [×3], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×4], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×3], S3×C2×C4 [×6], S3×C2×C4 [×8], C2×D12, S3×D4 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23 [×2], C2×C4×D4, S3×C42, C4×D12, S3×C22⋊C4 [×2], Dic3⋊4D4 [×2], S3×C4⋊C4, Dic3⋊5D4, C4×C3⋊D4 [×2], D4×Dic3, D4×C12, S3×C22×C4 [×2], C2×S3×D4, C4×S3×D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×4], C23 [×15], D6 [×7], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×S3 [×4], C22×S3 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, S3×C2×C4 [×6], S3×D4 [×2], S3×C23, C2×C4×D4, S3×C22×C4, C2×S3×D4, S3×C4○D4, C4×S3×D4
Generators and relations
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 >
►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 37 23)(2 38 24)(3 39 21)(4 40 22)(5 36 46)(6 33 47)(7 34 48)(8 35 45)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 43 17)(14 44 18)(15 41 19)(16 42 20)
(1 3)(2 4)(5 34)(6 35)(7 36)(8 33)(9 31)(10 32)(11 29)(12 30)(13 15)(14 16)(17 41)(18 42)(19 43)(20 44)(21 37)(22 38)(23 39)(24 40)(25 27)(26 28)(45 47)(46 48)
(1 45 27 16)(2 46 28 13)(3 47 25 14)(4 48 26 15)(5 10 43 38)(6 11 44 39)(7 12 41 40)(8 9 42 37)(17 24 36 30)(18 21 33 31)(19 22 34 32)(20 23 35 29)
(5 43)(6 44)(7 41)(8 42)(13 46)(14 47)(15 48)(16 45)(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), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,3)(2,4)(5,34)(6,35)(7,36)(8,33)(9,31)(10,32)(11,29)(12,30)(13,15)(14,16)(17,41)(18,42)(19,43)(20,44)(21,37)(22,38)(23,39)(24,40)(25,27)(26,28)(45,47)(46,48), (1,45,27,16)(2,46,28,13)(3,47,25,14)(4,48,26,15)(5,10,43,38)(6,11,44,39)(7,12,41,40)(8,9,42,37)(17,24,36,30)(18,21,33,31)(19,22,34,32)(20,23,35,29), (5,43)(6,44)(7,41)(8,42)(13,46)(14,47)(15,48)(16,45)(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), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,3)(2,4)(5,34)(6,35)(7,36)(8,33)(9,31)(10,32)(11,29)(12,30)(13,15)(14,16)(17,41)(18,42)(19,43)(20,44)(21,37)(22,38)(23,39)(24,40)(25,27)(26,28)(45,47)(46,48), (1,45,27,16)(2,46,28,13)(3,47,25,14)(4,48,26,15)(5,10,43,38)(6,11,44,39)(7,12,41,40)(8,9,42,37)(17,24,36,30)(18,21,33,31)(19,22,34,32)(20,23,35,29), (5,43)(6,44)(7,41)(8,42)(13,46)(14,47)(15,48)(16,45)(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)], [(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,36,46),(6,33,47),(7,34,48),(8,35,45),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,43,17),(14,44,18),(15,41,19),(16,42,20)], [(1,3),(2,4),(5,34),(6,35),(7,36),(8,33),(9,31),(10,32),(11,29),(12,30),(13,15),(14,16),(17,41),(18,42),(19,43),(20,44),(21,37),(22,38),(23,39),(24,40),(25,27),(26,28),(45,47),(46,48)], [(1,45,27,16),(2,46,28,13),(3,47,25,14),(4,48,26,15),(5,10,43,38),(6,11,44,39),(7,12,41,40),(8,9,42,37),(17,24,36,30),(18,21,33,31),(19,22,34,32),(20,23,35,29)], [(5,43),(6,44),(7,41),(8,42),(13,46),(14,47),(15,48),(16,45),(17,36),(18,33),(19,34),(20,35)])
Matrix representation ►G ⊆ 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] >;
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 |
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