direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C42⋊C2, C42⋊31D6, C4⋊C4⋊53D6, (S3×C42)⋊16C2, (C4×C12)⋊30C22, (C2×C6).64C24, C6.16(C23×C4), C42⋊2S3⋊28C2, D6.34(C4○D4), C4⋊Dic3⋊71C22, C22⋊C4.124D6, D6.23(C22×C4), (C22×C4).379D6, C12.142(C22×C4), (C2×C12).582C23, Dic3⋊C4⋊62C22, D6⋊C4.117C22, (C4×Dic3)⋊77C22, C22.26(S3×C23), C23.26D6⋊23C2, C23.16D6⋊30C2, (C22×C6).134C23, C23.162(C22×S3), Dic3.18(C22×C4), (S3×C23).103C22, (C22×S3).161C23, (C22×C12).224C22, (C2×Dic3).194C23, C6.D4.93C22, (C22×Dic3).215C22, (S3×C2×C4)⋊7C4, C4.93(S3×C2×C4), (S3×C4⋊C4)⋊46C2, (C2×C4)⋊16(C4×S3), C2.1(S3×C4○D4), (C4×S3)⋊14(C2×C4), (C2×C12)⋊10(C2×C4), C3⋊4(C2×C42⋊C2), C4⋊C4⋊7S3⋊45C2, (S3×C22×C4).6C2, C22.26(S3×C2×C4), C2.18(S3×C22×C4), (C3×C4⋊C4)⋊50C22, C6.131(C2×C4○D4), (S3×C22⋊C4).4C2, (C3×C42⋊C2)⋊6C2, (C2×Dic3)⋊23(C2×C4), (S3×C2×C4).245C22, (C2×C6).20(C22×C4), (C22×S3).66(C2×C4), (C2×C4).270(C22×S3), (C3×C22⋊C4).134C22, SmallGroup(192,1079)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C42⋊C2
G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, de=ed >
Subgroups: 712 in 330 conjugacy classes, 159 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, 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, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C23×C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C42⋊C2, S3×C42, C42⋊2S3, C23.16D6, S3×C22⋊C4, S3×C4⋊C4, C4⋊C4⋊7S3, C23.26D6, C3×C42⋊C2, S3×C22×C4, S3×C42⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C42⋊C2, C23×C4, C2×C4○D4, S3×C2×C4, S3×C23, C2×C42⋊C2, S3×C22×C4, S3×C4○D4, S3×C42⋊C2
►On 48 points
(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 27)(2 28)(3 25)(4 26)(5 17)(6 18)(7 19)(8 20)(9 23)(10 24)(11 21)(12 22)(13 46)(14 47)(15 48)(16 45)(29 37)(30 38)(31 39)(32 40)(33 44)(34 41)(35 42)(36 43)
(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 14 27 47)(2 15 28 48)(3 16 25 45)(4 13 26 46)(5 40 43 12)(6 37 44 9)(7 38 41 10)(8 39 42 11)(17 32 36 22)(18 29 33 23)(19 30 34 24)(20 31 35 21)
(1 3)(2 26)(4 28)(5 41)(6 8)(7 43)(9 11)(10 40)(12 38)(13 48)(14 16)(15 46)(17 34)(18 20)(19 36)(21 23)(22 30)(24 32)(25 27)(29 31)(33 35)(37 39)(42 44)(45 47)
G:=sub<Sym(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), (1,27)(2,28)(3,25)(4,26)(5,17)(6,18)(7,19)(8,20)(9,23)(10,24)(11,21)(12,22)(13,46)(14,47)(15,48)(16,45)(29,37)(30,38)(31,39)(32,40)(33,44)(34,41)(35,42)(36,43), (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), (1,14,27,47)(2,15,28,48)(3,16,25,45)(4,13,26,46)(5,40,43,12)(6,37,44,9)(7,38,41,10)(8,39,42,11)(17,32,36,22)(18,29,33,23)(19,30,34,24)(20,31,35,21), (1,3)(2,26)(4,28)(5,41)(6,8)(7,43)(9,11)(10,40)(12,38)(13,48)(14,16)(15,46)(17,34)(18,20)(19,36)(21,23)(22,30)(24,32)(25,27)(29,31)(33,35)(37,39)(42,44)(45,47)>;
G:=Group( (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), (1,27)(2,28)(3,25)(4,26)(5,17)(6,18)(7,19)(8,20)(9,23)(10,24)(11,21)(12,22)(13,46)(14,47)(15,48)(16,45)(29,37)(30,38)(31,39)(32,40)(33,44)(34,41)(35,42)(36,43), (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), (1,14,27,47)(2,15,28,48)(3,16,25,45)(4,13,26,46)(5,40,43,12)(6,37,44,9)(7,38,41,10)(8,39,42,11)(17,32,36,22)(18,29,33,23)(19,30,34,24)(20,31,35,21), (1,3)(2,26)(4,28)(5,41)(6,8)(7,43)(9,11)(10,40)(12,38)(13,48)(14,16)(15,46)(17,34)(18,20)(19,36)(21,23)(22,30)(24,32)(25,27)(29,31)(33,35)(37,39)(42,44)(45,47) );
G=PermutationGroup([[(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)], [(1,27),(2,28),(3,25),(4,26),(5,17),(6,18),(7,19),(8,20),(9,23),(10,24),(11,21),(12,22),(13,46),(14,47),(15,48),(16,45),(29,37),(30,38),(31,39),(32,40),(33,44),(34,41),(35,42),(36,43)], [(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)], [(1,14,27,47),(2,15,28,48),(3,16,25,45),(4,13,26,46),(5,40,43,12),(6,37,44,9),(7,38,41,10),(8,39,42,11),(17,32,36,22),(18,29,33,23),(19,30,34,24),(20,31,35,21)], [(1,3),(2,26),(4,28),(5,41),(6,8),(7,43),(9,11),(10,40),(12,38),(13,48),(14,16),(15,46),(17,34),(18,20),(19,36),(21,23),(22,30),(24,32),(25,27),(29,31),(33,35),(37,39),(42,44),(45,47)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 4S | ··· | 4AB | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4×S3 | S3×C4○D4 |
| kernel | S3×C42⋊C2 | S3×C42 | C42⋊2S3 | C23.16D6 | S3×C22⋊C4 | S3×C4⋊C4 | C4⋊C4⋊7S3 | C23.26D6 | C3×C42⋊C2 | S3×C22×C4 | S3×C2×C4 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | D6 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 8 | 4 |
Matrix representation of S3×C42⋊C2 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 12 | 0 | 0 |
| 0 | 2 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0],[5,0,0,0,0,0,1,2,0,0,0,12,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
S3×C42⋊C2 in GAP, Magma, Sage, TeX
S_3\times C_4^2\rtimes C_2
% in TeX
G:=Group("S3xC4^2:C2"); // GroupNames label
G:=SmallGroup(192,1079);
// by ID
G=gap.SmallGroup(192,1079);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^2=1,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=c*d^2,d*e=e*d>;
// generators/relations