direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4.4D4, C42⋊35D6, (C2×Q8)⋊20D6, C4.31(S3×D4), (S3×C42)⋊9C2, C22⋊C4⋊32D6, (C4×S3).24D4, D6.60(C2×D4), C12.60(C2×D4), (C4×C12)⋊21C22, D6⋊C4⋊29C22, (C2×D4).170D6, Dic3.9(C2×D4), (C6×Q8)⋊11C22, C6.87(C22×D4), C42⋊7S3⋊23C2, D6.40(C4○D4), (C2×C12).79C23, (C2×C6).217C24, C23.12D6⋊23C2, C12.23D4⋊20C2, (C4×Dic3)⋊79C22, (C2×Dic6)⋊32C22, (C6×D4).152C22, C23.49(C22×S3), (C22×C6).47C23, C23.11D6⋊39C2, (C2×D12).162C22, C6.D4⋊32C22, (C22×S3).95C23, (S3×C23).62C22, C22.238(S3×C23), (C2×Dic3).112C23, (C2×S3×Q8)⋊9C2, (C2×S3×D4).9C2, C2.60(C2×S3×D4), C3⋊3(C2×C4.4D4), C2.75(S3×C4○D4), (S3×C22⋊C4)⋊16C2, C6.186(C2×C4○D4), (C3×C4.4D4)⋊9C2, (S3×C2×C4).297C22, (C3×C22⋊C4)⋊27C22, (C2×C4).300(C22×S3), (C2×C3⋊D4).58C22, SmallGroup(192,1232)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 944 in 330 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×10], C22, C22 [×26], S3 [×4], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×17], D4 [×8], Q8 [×8], C23 [×2], C23 [×15], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], D6 [×14], C2×C6, C2×C6 [×6], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×12], C22×C4 [×5], C2×D4, C2×D4 [×7], C2×Q8, C2×Q8 [×7], C24 [×2], Dic6 [×6], C4×S3 [×4], C4×S3 [×8], D12 [×2], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×2], C22×S3 [×12], C22×C6 [×2], C2×C42, C2×C22⋊C4 [×4], C4.4D4, C4.4D4 [×7], C22×D4, C22×Q8, C4×Dic3, C4×Dic3 [×2], D6⋊C4 [×8], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12, S3×D4 [×4], S3×Q8 [×4], C2×C3⋊D4 [×2], C6×D4, C6×Q8, S3×C23 [×2], C2×C4.4D4, S3×C42, C42⋊7S3, S3×C22⋊C4 [×4], C23.11D6 [×4], C23.12D6, C12.23D4, C3×C4.4D4, C2×S3×D4, C2×S3×Q8, S3×C4.4D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C22×S3 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], S3×D4 [×2], S3×C23, C2×C4.4D4, C2×S3×D4, S3×C4○D4 [×2], S3×C4.4D4
Generators and relations
G = < a,b,c,d,e | a3=b2=c4=d4=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >
►On 48 points
(1 30 45)(2 31 46)(3 32 47)(4 29 48)(5 21 35)(6 22 36)(7 23 33)(8 24 34)(9 15 38)(10 16 39)(11 13 40)(12 14 37)(17 28 42)(18 25 43)(19 26 44)(20 27 41)
(1 40)(2 37)(3 38)(4 39)(5 43)(6 44)(7 41)(8 42)(9 47)(10 48)(11 45)(12 46)(13 30)(14 31)(15 32)(16 29)(17 34)(18 35)(19 36)(20 33)(21 25)(22 26)(23 27)(24 28)
(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 17 40 34)(2 18 37 35)(3 19 38 36)(4 20 39 33)(5 31 25 12)(6 32 26 9)(7 29 27 10)(8 30 28 11)(13 24 45 42)(14 21 46 43)(15 22 47 44)(16 23 48 41)
(1 35 3 33)(2 34 4 36)(5 32 7 30)(6 31 8 29)(9 27 11 25)(10 26 12 28)(13 43 15 41)(14 42 16 44)(17 39 19 37)(18 38 20 40)(21 47 23 45)(22 46 24 48)
G:=sub<Sym(48)| (1,30,45)(2,31,46)(3,32,47)(4,29,48)(5,21,35)(6,22,36)(7,23,33)(8,24,34)(9,15,38)(10,16,39)(11,13,40)(12,14,37)(17,28,42)(18,25,43)(19,26,44)(20,27,41), (1,40)(2,37)(3,38)(4,39)(5,43)(6,44)(7,41)(8,42)(9,47)(10,48)(11,45)(12,46)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,25)(22,26)(23,27)(24,28), (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), (1,17,40,34)(2,18,37,35)(3,19,38,36)(4,20,39,33)(5,31,25,12)(6,32,26,9)(7,29,27,10)(8,30,28,11)(13,24,45,42)(14,21,46,43)(15,22,47,44)(16,23,48,41), (1,35,3,33)(2,34,4,36)(5,32,7,30)(6,31,8,29)(9,27,11,25)(10,26,12,28)(13,43,15,41)(14,42,16,44)(17,39,19,37)(18,38,20,40)(21,47,23,45)(22,46,24,48)>;
G:=Group( (1,30,45)(2,31,46)(3,32,47)(4,29,48)(5,21,35)(6,22,36)(7,23,33)(8,24,34)(9,15,38)(10,16,39)(11,13,40)(12,14,37)(17,28,42)(18,25,43)(19,26,44)(20,27,41), (1,40)(2,37)(3,38)(4,39)(5,43)(6,44)(7,41)(8,42)(9,47)(10,48)(11,45)(12,46)(13,30)(14,31)(15,32)(16,29)(17,34)(18,35)(19,36)(20,33)(21,25)(22,26)(23,27)(24,28), (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), (1,17,40,34)(2,18,37,35)(3,19,38,36)(4,20,39,33)(5,31,25,12)(6,32,26,9)(7,29,27,10)(8,30,28,11)(13,24,45,42)(14,21,46,43)(15,22,47,44)(16,23,48,41), (1,35,3,33)(2,34,4,36)(5,32,7,30)(6,31,8,29)(9,27,11,25)(10,26,12,28)(13,43,15,41)(14,42,16,44)(17,39,19,37)(18,38,20,40)(21,47,23,45)(22,46,24,48) );
G=PermutationGroup([(1,30,45),(2,31,46),(3,32,47),(4,29,48),(5,21,35),(6,22,36),(7,23,33),(8,24,34),(9,15,38),(10,16,39),(11,13,40),(12,14,37),(17,28,42),(18,25,43),(19,26,44),(20,27,41)], [(1,40),(2,37),(3,38),(4,39),(5,43),(6,44),(7,41),(8,42),(9,47),(10,48),(11,45),(12,46),(13,30),(14,31),(15,32),(16,29),(17,34),(18,35),(19,36),(20,33),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,17,40,34),(2,18,37,35),(3,19,38,36),(4,20,39,33),(5,31,25,12),(6,32,26,9),(7,29,27,10),(8,30,28,11),(13,24,45,42),(14,21,46,43),(15,22,47,44),(16,23,48,41)], [(1,35,3,33),(2,34,4,36),(5,32,7,30),(6,31,8,29),(9,27,11,25),(10,26,12,28),(13,43,15,41),(14,42,16,44),(17,39,19,37),(18,38,20,40),(21,47,23,45),(22,46,24,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,1,0,0,0,0,2,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,12,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | S3×D4 | S3×C4○D4 |
| kernel | S3×C4.4D4 | S3×C42 | C42⋊7S3 | S3×C22⋊C4 | C23.11D6 | C23.12D6 | C12.23D4 | C3×C4.4D4 | C2×S3×D4 | C2×S3×Q8 | C4.4D4 | C4×S3 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
S_3\times C_4._4D_4
% in TeX
G:=Group("S3xC4.4D4"); // GroupNames label
G:=SmallGroup(192,1232);
// by ID
G=gap.SmallGroup(192,1232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,346,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=1,e^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=c^2*d^-1>;
// generators/relations