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, (C2×C12).207D4, C12.249(C2×D4), (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
Subgroups: 744 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×2], C6, C6 [×2], C6 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×18], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×10], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×8], D4.S3 [×8], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C6×D4 [×6], C6×D4 [×3], C23×C6, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3 [×2], D12⋊6C22 [×8], C2×D4.S3 [×2], C2×C4○D12, D4×C2×C6, C2×D12⋊6C22
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8⋊C22, D12⋊6C22 [×2], C22×C3⋊D4, C2×D12⋊6C22
Generators and relations
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 >
►On 48 points
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 41)(14 40)(15 39)(16 38)(17 37)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 24)(2 19)(3 14)(4 21)(5 16)(6 23)(7 18)(8 13)(9 20)(10 15)(11 22)(12 17)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)
G:=sub<Sym(48)| (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,24)(2,19)(3,14)(4,21)(5,16)(6,23)(7,18)(8,13)(9,20)(10,15)(11,22)(12,17)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45)>;
G:=Group( (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,24)(2,19)(3,14)(4,21)(5,16)(6,23)(7,18)(8,13)(9,20)(10,15)(11,22)(12,17)(25,40)(26,47)(27,42)(28,37)(29,44)(30,39)(31,46)(32,41)(33,48)(34,43)(35,38)(36,45) );
G=PermutationGroup([(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,41),(14,40),(15,39),(16,38),(17,37),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,13),(9,14),(10,15),(11,16),(12,17),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,24),(2,19),(3,14),(4,21),(5,16),(6,23),(7,18),(8,13),(9,20),(10,15),(11,22),(12,17),(25,40),(26,47),(27,42),(28,37),(29,44),(30,39),(31,46),(32,41),(33,48),(34,43),(35,38),(36,45)])
Matrix representation ►G ⊆ 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] >;
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 |
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