direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic3, C6⋊2M4(2), C12.41C23, C23.4Dic3, C3⋊C8⋊12C22, (C2×C12).9C4, C3⋊3(C2×M4(2)), C12.36(C2×C4), (C2×C4).100D6, C4○(C4.Dic3), (C22×C4).6S3, (C22×C6).7C4, (C2×C4).6Dic3, C4.9(C2×Dic3), (C22×C12).9C2, C6.21(C22×C4), C4.41(C22×S3), C2.3(C22×Dic3), (C2×C12).100C22, C22.12(C2×Dic3), (C2×C3⋊C8)⋊12C2, (C2×C6).32(C2×C4), SmallGroup(96,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic3
G = < a,b,c,d | a2=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 98 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C2×M4(2), C2×C3⋊C8, C4.Dic3, C22×C12, C2×C4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C2×Dic3, C22×S3, C2×M4(2), C4.Dic3, C22×Dic3, C2×C4.Dic3
►On 48 points
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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 36 10 33 7 30 4 27)(2 29 11 26 8 35 5 32)(3 34 12 31 9 28 6 25)(13 41 22 38 19 47 16 44)(14 46 23 43 20 40 17 37)(15 39 24 48 21 45 18 42)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,36,10,33,7,30,4,27)(2,29,11,26,8,35,5,32)(3,34,12,31,9,28,6,25)(13,41,22,38,19,47,16,44)(14,46,23,43,20,40,17,37)(15,39,24,48,21,45,18,42)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,36,10,33,7,30,4,27)(2,29,11,26,8,35,5,32)(3,34,12,31,9,28,6,25)(13,41,22,38,19,47,16,44)(14,46,23,43,20,40,17,37)(15,39,24,48,21,45,18,42) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(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)], [(1,36,10,33,7,30,4,27),(2,29,11,26,8,35,5,32),(3,34,12,31,9,28,6,25),(13,41,22,38,19,47,16,44),(14,46,23,43,20,40,17,37),(15,39,24,48,21,45,18,42)]])
C2×C4.Dic3 is a maximal subgroup of
C12.8C42 C12.(C4⋊C4) C42⋊3Dic3 C12.2C42 (C2×C12).Q8 C12.10C42 M4(2)⋊Dic3 (C2×C24)⋊C4 C12.4C42 M4(2)⋊4Dic3 C12.21C42 D6⋊2M4(2) Dic3⋊M4(2) C12⋊7M4(2) C4⋊C4.225D6 C4○D12⋊C4 C4⋊C4.232D6 C12.5C42 C42.43D6 C4⋊C4⋊36D6 C4⋊C4.237D6 C42⋊6D6 C42.47D6 C12⋊3M4(2) C4⋊D4⋊S3 C3⋊C8⋊5D4 C3⋊C8⋊6D4 C3⋊C8.6D4 C12.12C42 Dic3⋊C8⋊C2 (C22×C8)⋊7S3 Dic3×M4(2) Dic3⋊4M4(2) C23.8Dic6 C23.9Dic6 D6⋊6M4(2) M4(2).31D6 C24.6Dic3 (C6×D4)⋊6C4 (C6×Q8)⋊6C4 C4○D4⋊3Dic3 (C6×D4).11C4 (C6×D4)⋊9C4 (C6×D4).16C4 C2×S3×M4(2) M4(2)⋊26D6 C12.76C24 C12.C24 C60.59(C2×C4)
C2×C4.Dic3 is a maximal quotient of
C12⋊7M4(2) C42.285D6 C42.270D6 C42.47D6 C12⋊3M4(2) C42.210D6 C24.6Dic3 C60.59(C2×C4)
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 |
| kernel | C2×C4.Dic3 | C2×C3⋊C8 | C4.Dic3 | C22×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 1 | 3 | 3 | 1 | 4 | 8 |
Matrix representation of C2×C4.Dic3 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 27 |
| 72 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 24 |
| 46 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 46 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,46,0,0,0,27],[72,0,0,0,3,0,0,0,24],[46,0,0,0,0,46,0,1,0] >;
C2×C4.Dic3 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C2xC4.Dic3"); // GroupNames label
G:=SmallGroup(96,128);
// by ID
G=gap.SmallGroup(96,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations