metabelian, supersoluble, monomial
Aliases: C24⋊2S3, C6.7D12, C12.46D6, C32⋊7SD16, C8⋊2(C3⋊S3), (C3×C24)⋊3C2, (C3×C6).22D4, C3⋊1(C24⋊C2), C32⋊4Q8⋊1C2, C12⋊S3.1C2, C2.3(C12⋊S3), (C3×C12).32C22, C4.8(C2×C3⋊S3), SmallGroup(144,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2S3
G = < a,b,c | a24=b3=c2=1, ab=ba, cac=a11, cbc=b-1 >
Subgroups: 266 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, D4, Q8, C32, Dic3 [×4], C12 [×4], D6 [×4], SD16, C3⋊S3, C3×C6, C24 [×4], Dic6 [×4], D12 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, C24⋊C2 [×4], C3×C24, C32⋊4Q8, C12⋊S3, C24⋊2S3
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], SD16, C3⋊S3, D12 [×4], C2×C3⋊S3, C24⋊C2 [×4], C12⋊S3, C24⋊2S3
►On 72 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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 38 72)(2 39 49)(3 40 50)(4 41 51)(5 42 52)(6 43 53)(7 44 54)(8 45 55)(9 46 56)(10 47 57)(11 48 58)(12 25 59)(13 26 60)(14 27 61)(15 28 62)(16 29 63)(17 30 64)(18 31 65)(19 32 66)(20 33 67)(21 34 68)(22 35 69)(23 36 70)(24 37 71)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 49)(26 60)(27 71)(28 58)(29 69)(30 56)(31 67)(32 54)(33 65)(34 52)(35 63)(36 50)(37 61)(38 72)(39 59)(40 70)(41 57)(42 68)(43 55)(44 66)(45 53)(46 64)(47 51)(48 62)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,38,72)(2,39,49)(3,40,50)(4,41,51)(5,42,52)(6,43,53)(7,44,54)(8,45,55)(9,46,56)(10,47,57)(11,48,58)(12,25,59)(13,26,60)(14,27,61)(15,28,62)(16,29,63)(17,30,64)(18,31,65)(19,32,66)(20,33,67)(21,34,68)(22,35,69)(23,36,70)(24,37,71), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,49)(26,60)(27,71)(28,58)(29,69)(30,56)(31,67)(32,54)(33,65)(34,52)(35,63)(36,50)(37,61)(38,72)(39,59)(40,70)(41,57)(42,68)(43,55)(44,66)(45,53)(46,64)(47,51)(48,62)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,38,72)(2,39,49)(3,40,50)(4,41,51)(5,42,52)(6,43,53)(7,44,54)(8,45,55)(9,46,56)(10,47,57)(11,48,58)(12,25,59)(13,26,60)(14,27,61)(15,28,62)(16,29,63)(17,30,64)(18,31,65)(19,32,66)(20,33,67)(21,34,68)(22,35,69)(23,36,70)(24,37,71), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,49)(26,60)(27,71)(28,58)(29,69)(30,56)(31,67)(32,54)(33,65)(34,52)(35,63)(36,50)(37,61)(38,72)(39,59)(40,70)(41,57)(42,68)(43,55)(44,66)(45,53)(46,64)(47,51)(48,62) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,38,72),(2,39,49),(3,40,50),(4,41,51),(5,42,52),(6,43,53),(7,44,54),(8,45,55),(9,46,56),(10,47,57),(11,48,58),(12,25,59),(13,26,60),(14,27,61),(15,28,62),(16,29,63),(17,30,64),(18,31,65),(19,32,66),(20,33,67),(21,34,68),(22,35,69),(23,36,70),(24,37,71)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,49),(26,60),(27,71),(28,58),(29,69),(30,56),(31,67),(32,54),(33,65),(34,52),(35,63),(36,50),(37,61),(38,72),(39,59),(40,70),(41,57),(42,68),(43,55),(44,66),(45,53),(46,64),(47,51),(48,62)])
C24⋊2S3 is a maximal subgroup of
S3×C24⋊C2 D24⋊S3 Dic12⋊S3 D6.1D12 C24.78D6 C24⋊3D6 C24.5D6 C24⋊8D6 SD16×C3⋊S3 C24.40D6 C24.35D6 He3⋊6SD16 C24⋊D9 C33⋊16SD16 C33⋊17SD16 C33⋊21SD16
C24⋊2S3 is a maximal quotient of
C6.4Dic12 C24⋊2Dic3 C62.84D4 C24⋊D9 He3⋊7SD16 C33⋊16SD16 C33⋊17SD16 C33⋊21SD16
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 2 | 2 | 2 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | SD16 | D12 | C24⋊C2 |
| kernel | C24⋊2S3 | C3×C24 | C32⋊4Q8 | C12⋊S3 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 16 |
Matrix representation of C24⋊2S3 ►in GL4(𝔽73) generated by
| 62 | 48 | 0 | 0 |
| 25 | 37 | 0 | 0 |
| 0 | 0 | 37 | 48 |
| 0 | 0 | 25 | 62 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [62,25,0,0,48,37,0,0,0,0,37,25,0,0,48,62],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,72,0] >;
C24⋊2S3 in GAP, Magma, Sage, TeX
C_{24}\rtimes_2S_3 % in TeX
G:=Group("C24:2S3"); // GroupNames label
G:=SmallGroup(144,87);
// by ID
G=gap.SmallGroup(144,87);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,31,218,50,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations