metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6×D4).16C4, (C2×D4).203D6, (C2×C12).199D4, C12.214(C2×D4), (C22×C12).8C4, (C2×Q8).196D6, (C2×D4).9Dic3, C12.D4⋊13C2, (C22×C4).179D6, C12.10D4⋊13C2, C23.8(C2×Dic3), (C22×C4).9Dic3, C12.91(C22⋊C4), (C2×C12).483C23, (C6×D4).244C22, (C6×Q8).207C22, C4.34(C6.D4), C4.Dic3.47C22, C22.8(C22×Dic3), (C22×C12).209C22, C22.7(C6.D4), C3⋊4(M4(2).8C22), (C6×C4○D4).6C2, C4.96(C2×C3⋊D4), (C2×C12).16(C2×C4), (C2×C4○D4).12S3, C6.86(C2×C22⋊C4), (C2×C4).5(C2×Dic3), (C2×C4).91(C3⋊D4), (C2×C4.Dic3)⋊23C2, (C22×C6).75(C2×C4), (C2×C6).28(C22⋊C4), (C2×C6).201(C22×C4), (C2×C4).131(C22×S3), C2.22(C2×C6.D4), SmallGroup(192,796)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×D4).16C4
G = < a,b,c,d | a6=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a3b, dcd-1=b2c >
Subgroups: 296 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, M4(2).8C22, C12.D4, C12.10D4, C2×C4.Dic3, C6×C4○D4, (C6×D4).16C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, M4(2).8C22, C2×C6.D4, (C6×D4).16C4
►On 48 points
(1 22 28 5 18 32)(2 29 19)(3 24 30 7 20 26)(4 31 21)(6 25 23)(8 27 17)(9 39 43 13 35 47)(10 44 36)(11 33 45 15 37 41)(12 46 38)(14 48 40)(16 42 34)
(1 13 5 9)(2 10 6 14)(3 11 7 15)(4 16 8 12)(17 38 21 34)(18 39 22 35)(19 36 23 40)(20 37 24 33)(25 48 29 44)(26 41 30 45)(27 46 31 42)(28 47 32 43)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 36)(18 33)(19 38)(20 35)(21 40)(22 37)(23 34)(24 39)(25 42)(26 47)(27 44)(28 41)(29 46)(30 43)(31 48)(32 45)
(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)
G:=sub<Sym(48)| (1,22,28,5,18,32)(2,29,19)(3,24,30,7,20,26)(4,31,21)(6,25,23)(8,27,17)(9,39,43,13,35,47)(10,44,36)(11,33,45,15,37,41)(12,46,38)(14,48,40)(16,42,34), (1,13,5,9)(2,10,6,14)(3,11,7,15)(4,16,8,12)(17,38,21,34)(18,39,22,35)(19,36,23,40)(20,37,24,33)(25,48,29,44)(26,41,30,45)(27,46,31,42)(28,47,32,43), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,36)(18,33)(19,38)(20,35)(21,40)(22,37)(23,34)(24,39)(25,42)(26,47)(27,44)(28,41)(29,46)(30,43)(31,48)(32,45), (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)>;
G:=Group( (1,22,28,5,18,32)(2,29,19)(3,24,30,7,20,26)(4,31,21)(6,25,23)(8,27,17)(9,39,43,13,35,47)(10,44,36)(11,33,45,15,37,41)(12,46,38)(14,48,40)(16,42,34), (1,13,5,9)(2,10,6,14)(3,11,7,15)(4,16,8,12)(17,38,21,34)(18,39,22,35)(19,36,23,40)(20,37,24,33)(25,48,29,44)(26,41,30,45)(27,46,31,42)(28,47,32,43), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,36)(18,33)(19,38)(20,35)(21,40)(22,37)(23,34)(24,39)(25,42)(26,47)(27,44)(28,41)(29,46)(30,43)(31,48)(32,45), (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) );
G=PermutationGroup([[(1,22,28,5,18,32),(2,29,19),(3,24,30,7,20,26),(4,31,21),(6,25,23),(8,27,17),(9,39,43,13,35,47),(10,44,36),(11,33,45,15,37,41),(12,46,38),(14,48,40),(16,42,34)], [(1,13,5,9),(2,10,6,14),(3,11,7,15),(4,16,8,12),(17,38,21,34),(18,39,22,35),(19,36,23,40),(20,37,24,33),(25,48,29,44),(26,41,30,45),(27,46,31,42),(28,47,32,43)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10),(17,36),(18,33),(19,38),(20,35),(21,40),(22,37),(23,34),(24,39),(25,42),(26,47),(27,44),(28,41),(29,46),(30,43),(31,48),(32,45)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | ··· | 6I | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Dic3 | D6 | Dic3 | D6 | D6 | C3⋊D4 | M4(2).8C22 | (C6×D4).16C4 |
| kernel | (C6×D4).16C4 | C12.D4 | C12.10D4 | C2×C4.Dic3 | C6×C4○D4 | C22×C12 | C6×D4 | C2×C4○D4 | C2×C12 | C22×C4 | C22×C4 | C2×D4 | C2×D4 | C2×Q8 | C2×C4 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 4 |
Matrix representation of (C6×D4).16C4 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 8 |
| 0 | 9 | 0 | 65 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 0 | 0 | 6 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 46 |
| 0 | 0 | 0 | 27 |
| 0 | 46 | 6 | 0 |
| 27 | 0 | 67 | 67 |
| 0 | 0 | 46 | 46 |
| 0 | 0 | 54 | 27 |
| 57 | 0 | 1 | 55 |
| 16 | 0 | 0 | 18 |
| 72 | 1 | 0 | 57 |
| 2 | 0 | 0 | 16 |
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,8,0,8,65,0,8],[46,0,0,0,0,27,0,0,0,0,46,0,6,0,46,27],[0,27,0,0,46,0,0,0,6,67,46,54,0,67,46,27],[57,16,72,2,0,0,1,0,1,0,0,0,55,18,57,16] >;
(C6×D4).16C4 in GAP, Magma, Sage, TeX
(C_6\times D_4)._{16}C_4 % in TeX
G:=Group("(C6xD4).16C4"); // GroupNames label
G:=SmallGroup(192,796);
// by ID
G=gap.SmallGroup(192,796);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,d*b*d^-1=a^3*b,d*c*d^-1=b^2*c>;
// generators/relations