metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.35C24, D12.31C23, Dic6.30C23, C4○D4⋊19D6, (C2×D4)⋊42D6, (C2×Q8)⋊34D6, C3⋊C8.14C23, D4⋊D6⋊13C2, D4⋊S3⋊19C22, Q8.13D6⋊7C2, (C2×C12).218D4, C12.427(C2×D4), (C6×D4)⋊46C22, C4.35(S3×C23), (C6×Q8)⋊38C22, Q8.14D6⋊13C2, D12⋊6C22⋊13C2, C4○D12⋊21C22, (C2×D12)⋊59C22, C3⋊5(D8⋊C22), D4.S3⋊17C22, C3⋊Q16⋊16C22, (C3×D4).23C23, D4.23(C22×S3), (C22×C6).124D4, (C22×C4).298D6, C6.160(C22×D4), (C3×Q8).23C23, Q8.33(C22×S3), Q8.11D6⋊13C2, (C2×C12).557C23, Q8⋊2S3⋊18C22, (C2×Dic6)⋊69C22, C23.41(C3⋊D4), C4.Dic3⋊37C22, (C22×C12).292C22, (C6×C4○D4)⋊4C2, (C2×C4○D4)⋊8S3, (C2×C3⋊C8)⋊23C22, (C2×C4○D12)⋊31C2, (C2×C6).591(C2×D4), C4.121(C2×C3⋊D4), (C3×C4○D4)⋊18C22, (C2×C4.Dic3)⋊31C2, C22.21(C2×C3⋊D4), C2.33(C22×C3⋊D4), (C2×C4).203(C3⋊D4), (C2×C4).246(C22×S3), SmallGroup(192,1381)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.C24
G = < a,b,c,d,e | a12=b2=c2=e2=1, d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a9b, cd=dc, ece=a6c, de=ed >
Subgroups: 616 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, D8⋊C22, C2×C4.Dic3, D12⋊6C22, Q8.11D6, D4⋊D6, Q8.13D6, Q8.14D6, C2×C4○D12, C6×C4○D4, C12.C24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C3⋊D4, S3×C23, D8⋊C22, C22×C3⋊D4, C12.C24
►On 48 points
(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)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(26 36)(27 35)(28 34)(29 33)(30 32)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 37)(26 44)(27 39)(28 46)(29 41)(30 48)(31 43)(32 38)(33 45)(34 40)(35 47)(36 42)
G:=sub<Sym(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), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42)>;
G:=Group( (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), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42) );
G=PermutationGroup([[(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)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(26,36),(27,35),(28,34),(29,33),(30,32),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,43,19,37),(14,44,20,38),(15,45,21,39),(16,46,22,40),(17,47,23,41),(18,48,24,42)], [(1,13),(2,20),(3,15),(4,22),(5,17),(6,24),(7,19),(8,14),(9,21),(10,16),(11,23),(12,18),(25,37),(26,44),(27,39),(28,46),(29,41),(30,48),(31,43),(32,38),(33,45),(34,40),(35,47),(36,42)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | ··· | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | D8⋊C22 | C12.C24 |
| kernel | C12.C24 | C2×C4.Dic3 | D12⋊6C22 | Q8.11D6 | D4⋊D6 | Q8.13D6 | Q8.14D6 | C2×C4○D12 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 6 | 2 | 2 | 4 |
Matrix representation of C12.C24 ►in GL4(𝔽73) generated by
| 14 | 7 | 0 | 0 |
| 66 | 7 | 0 | 0 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 66 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 14 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,59,7,0,0,66,66],[0,72,0,0,72,0,0,0,0,0,59,7,0,0,66,14],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
C12.C24 in GAP, Magma, Sage, TeX
C_{12}.C_2^4 % in TeX
G:=Group("C12.C2^4"); // GroupNames label
G:=SmallGroup(192,1381);
// by ID
G=gap.SmallGroup(192,1381);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=c^2=e^2=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^7,b*c=c*b,b*d=d*b,e*b*e=a^9*b,c*d=d*c,e*c*e=a^6*c,d*e=e*d>;
// generators/relations