direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×C4○D4, C6.11C25, D12⋊11C23, C12.46C24, D6.13C24, Dic6⋊11C23, Dic3.6C24, (C2×D4)⋊49D6, (C2×Q8)⋊41D6, (C4×S3)⋊7C23, (C2×C12)⋊6C23, D4⋊8(C22×S3), (C3×D4)⋊9C23, (C22×C4)⋊43D6, C3⋊D4⋊4C23, (C2×C6).2C24, Q8⋊8(C22×S3), (C3×Q8)⋊8C23, (S3×D4)⋊16C22, (C6×D4)⋊52C22, C4.77(S3×C23), C2.12(S3×C24), (S3×Q8)⋊19C22, (C6×Q8)⋊45C22, C4○D12⋊24C22, (C2×D12)⋊63C22, C22.2(S3×C23), D4⋊2S3⋊18C22, (C22×C12)⋊27C22, Q8⋊3S3⋊18C22, (C2×Dic3)⋊12C23, (C2×Dic6)⋊74C22, C23.223(C22×S3), (C22×C6).247C23, (S3×C23).119C22, (C22×S3).249C23, (C22×Dic3)⋊53C22, (C2×S3×D4)⋊30C2, C6⋊4(C2×C4○D4), (C2×S3×Q8)⋊23C2, C3⋊4(C22×C4○D4), (C6×C4○D4)⋊13C2, (S3×C2×C4)⋊61C22, (S3×C22×C4)⋊11C2, (C2×C4)⋊9(C22×S3), (C2×C4○D12)⋊36C2, (C2×D4⋊2S3)⋊32C2, (C2×Q8⋊3S3)⋊23C2, (C3×C4○D4)⋊19C22, (C2×C3⋊D4)⋊53C22, SmallGroup(192,1520)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×S3×C4○D4
G = < a,b,c,d,e,f | a2=b3=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >
Subgroups: 1832 in 890 conjugacy classes, 455 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C22×C4○D4, S3×C22×C4, C2×C4○D12, C2×S3×D4, C2×D4⋊2S3, C2×S3×Q8, C2×Q8⋊3S3, S3×C4○D4, C6×C4○D4, C2×S3×C4○D4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C25, S3×C23, C22×C4○D4, S3×C4○D4, S3×C24, C2×S3×C4○D4
►On 48 points
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 36)(2 33)(3 34)(4 35)(5 32)(6 29)(7 30)(8 31)(9 26)(10 27)(11 28)(12 25)(13 44)(14 41)(15 42)(16 43)(17 38)(18 39)(19 40)(20 37)(21 47)(22 48)(23 45)(24 46)
(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 46 3 48)(2 47 4 45)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)(21 35 23 33)(22 36 24 34)(25 39 27 37)(26 40 28 38)(29 43 31 41)(30 44 32 42)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(45 47)(46 48)
G:=sub<Sym(48)| (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,39,44)(34,40,41)(35,37,42)(36,38,43), (1,36)(2,33)(3,34)(4,35)(5,32)(6,29)(7,30)(8,31)(9,26)(10,27)(11,28)(12,25)(13,44)(14,41)(15,42)(16,43)(17,38)(18,39)(19,40)(20,37)(21,47)(22,48)(23,45)(24,46), (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), (1,46,3,48)(2,47,4,45)(5,15,7,13)(6,16,8,14)(9,19,11,17)(10,20,12,18)(21,35,23,33)(22,36,24,34)(25,39,27,37)(26,40,28,38)(29,43,31,41)(30,44,32,42), (5,7)(6,8)(9,11)(10,12)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(45,47)(46,48)>;
G:=Group( (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,39,44)(34,40,41)(35,37,42)(36,38,43), (1,36)(2,33)(3,34)(4,35)(5,32)(6,29)(7,30)(8,31)(9,26)(10,27)(11,28)(12,25)(13,44)(14,41)(15,42)(16,43)(17,38)(18,39)(19,40)(20,37)(21,47)(22,48)(23,45)(24,46), (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), (1,46,3,48)(2,47,4,45)(5,15,7,13)(6,16,8,14)(9,19,11,17)(10,20,12,18)(21,35,23,33)(22,36,24,34)(25,39,27,37)(26,40,28,38)(29,43,31,41)(30,44,32,42), (5,7)(6,8)(9,11)(10,12)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(45,47)(46,48) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,33),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,47),(6,11,48),(7,12,45),(8,9,46),(21,27,32),(22,28,29),(23,25,30),(24,26,31),(33,39,44),(34,40,41),(35,37,42),(36,38,43)], [(1,36),(2,33),(3,34),(4,35),(5,32),(6,29),(7,30),(8,31),(9,26),(10,27),(11,28),(12,25),(13,44),(14,41),(15,42),(16,43),(17,38),(18,39),(19,40),(20,37),(21,47),(22,48),(23,45),(24,46)], [(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)], [(1,46,3,48),(2,47,4,45),(5,15,7,13),(6,16,8,14),(9,19,11,17),(10,20,12,18),(21,35,23,33),(22,36,24,34),(25,39,27,37),(26,40,28,38),(29,43,31,41),(30,44,32,42)], [(5,7),(6,8),(9,11),(10,12),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(45,47),(46,48)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 2N | ··· | 2S | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 4O | ··· | 4T | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | S3×C4○D4 |
| kernel | C2×S3×C4○D4 | S3×C22×C4 | C2×C4○D12 | C2×S3×D4 | C2×D4⋊2S3 | C2×S3×Q8 | C2×Q8⋊3S3 | S3×C4○D4 | C6×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | D6 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 16 | 1 | 1 | 3 | 3 | 1 | 8 | 8 | 4 |
Matrix representation of C2×S3×C4○D4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,11,12],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;
C2×S3×C4○D4 in GAP, Magma, Sage, TeX
C_2\times S_3\times C_4\circ D_4
% in TeX
G:=Group("C2xS3xC4oD4"); // GroupNames label
G:=SmallGroup(192,1520);
// by ID
G=gap.SmallGroup(192,1520);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^4=f^2=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d^2*e>;
// generators/relations