direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12⋊6C22, D12⋊7C23, C12.28C24, Dic6⋊6C23, C3⋊C8⋊4C23, (C2×D4)⋊34D6, C6⋊4(C8⋊C22), (C22×D4)⋊7S3, D4⋊S3⋊17C22, C12.249(C2×D4), (C2×C12).207D4, (C6×D4)⋊42C22, C4.28(S3×C23), C4○D12⋊19C22, (C2×D12)⋊55C22, D4.S3⋊16C22, D4.20(C22×S3), (C3×D4).20C23, (C22×C6).207D4, C6.137(C22×D4), (C22×C4).284D6, (C2×C12).537C23, (C2×Dic6)⋊65C22, C23.99(C3⋊D4), C4.Dic3⋊32C22, (C22×C12).270C22, (D4×C2×C6)⋊3C2, C3⋊5(C2×C8⋊C22), (C2×D4⋊S3)⋊30C2, (C2×C3⋊C8)⋊20C22, C4.21(C2×C3⋊D4), (C2×C4○D12)⋊28C2, (C2×D4.S3)⋊30C2, (C2×C6).577(C2×D4), (C2×C4).92(C3⋊D4), (C2×C4.Dic3)⋊26C2, C2.10(C22×C3⋊D4), (C2×C4).235(C22×S3), C22.106(C2×C3⋊D4), SmallGroup(192,1352)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D12⋊6C22
G = < a,b,c,d,e | a2=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >
Subgroups: 744 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C23×C6, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3, D12⋊6C22, C2×D4.S3, C2×C4○D12, D4×C2×C6, C2×D12⋊6C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C8⋊C22, C22×D4, C2×C3⋊D4, S3×C23, C2×C8⋊C22, D12⋊6C22, C22×C3⋊D4, C2×D12⋊6C22
►On 48 points
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(1 22)(2 17)(3 24)(4 19)(5 14)(6 21)(7 16)(8 23)(9 18)(10 13)(11 20)(12 15)(25 43)(26 38)(27 45)(28 40)(29 47)(30 42)(31 37)(32 44)(33 39)(34 46)(35 41)(36 48)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48)>;
G:=Group( (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15)(25,43)(26,38)(27,45)(28,40)(29,47)(30,42)(31,37)(32,44)(33,39)(34,46)(35,41)(36,48) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39)], [(1,22),(2,17),(3,24),(4,19),(5,14),(6,21),(7,16),(8,23),(9,18),(10,13),(11,20),(12,15),(25,43),(26,38),(27,45),(28,40),(29,47),(30,42),(31,37),(32,44),(33,39),(34,46),(35,41),(36,48)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 6H | ··· | 6O | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 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 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8⋊C22 | D12⋊6C22 |
| kernel | C2×D12⋊6C22 | C2×C4.Dic3 | C2×D4⋊S3 | D12⋊6C22 | C2×D4.S3 | C2×C4○D12 | D4×C2×C6 | C22×D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 1 | 3 | 1 | 1 | 6 | 6 | 2 | 2 | 4 |
Matrix representation of C2×D12⋊6C22 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 29 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 60 | 0 | 0 |
| 0 | 0 | 66 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 31 |
| 0 | 0 | 0 | 0 | 56 | 64 |
| 52 | 40 | 0 | 0 | 0 | 0 |
| 62 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 31 |
| 0 | 0 | 0 | 0 | 56 | 64 |
| 0 | 0 | 8 | 60 | 0 | 0 |
| 0 | 0 | 66 | 65 | 0 | 0 |
| 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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,29,0,0,0,0,0,64,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,0,0,0,0,9,56,0,0,0,0,31,64],[52,62,0,0,0,0,40,21,0,0,0,0,0,0,0,0,8,66,0,0,0,0,60,65,0,0,9,56,0,0,0,0,31,64,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,10,0,0,0,0,0,1] >;
C2×D12⋊6C22 in GAP, Magma, Sage, TeX
C_2\times D_{12}\rtimes_6C_2^2 % in TeX
G:=Group("C2xD12:6C2^2"); // GroupNames label
G:=SmallGroup(192,1352);
// by ID
G=gap.SmallGroup(192,1352);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations