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, C4, C4, C22, S3, C6, C8, D4, Q8, C32, Dic3, C12, D6, SD16, C3⋊S3, C3×C6, C24, Dic6, D12, C3⋊Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, C3×C24, C32⋊4Q8, C12⋊S3, C24⋊2S3
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, D12, C2×C3⋊S3, C24⋊C2, 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 65 32)(2 66 33)(3 67 34)(4 68 35)(5 69 36)(6 70 37)(7 71 38)(8 72 39)(9 49 40)(10 50 41)(11 51 42)(12 52 43)(13 53 44)(14 54 45)(15 55 46)(16 56 47)(17 57 48)(18 58 25)(19 59 26)(20 60 27)(21 61 28)(22 62 29)(23 63 30)(24 64 31)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 60)(26 71)(27 58)(28 69)(29 56)(30 67)(31 54)(32 65)(33 52)(34 63)(35 50)(36 61)(37 72)(38 59)(39 70)(40 57)(41 68)(42 55)(43 66)(44 53)(45 64)(46 51)(47 62)(48 49)
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), (1,65,32)(2,66,33)(3,67,34)(4,68,35)(5,69,36)(6,70,37)(7,71,38)(8,72,39)(9,49,40)(10,50,41)(11,51,42)(12,52,43)(13,53,44)(14,54,45)(15,55,46)(16,56,47)(17,57,48)(18,58,25)(19,59,26)(20,60,27)(21,61,28)(22,62,29)(23,63,30)(24,64,31), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,60)(26,71)(27,58)(28,69)(29,56)(30,67)(31,54)(32,65)(33,52)(34,63)(35,50)(36,61)(37,72)(38,59)(39,70)(40,57)(41,68)(42,55)(43,66)(44,53)(45,64)(46,51)(47,62)(48,49)>;
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), (1,65,32)(2,66,33)(3,67,34)(4,68,35)(5,69,36)(6,70,37)(7,71,38)(8,72,39)(9,49,40)(10,50,41)(11,51,42)(12,52,43)(13,53,44)(14,54,45)(15,55,46)(16,56,47)(17,57,48)(18,58,25)(19,59,26)(20,60,27)(21,61,28)(22,62,29)(23,63,30)(24,64,31), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,60)(26,71)(27,58)(28,69)(29,56)(30,67)(31,54)(32,65)(33,52)(34,63)(35,50)(36,61)(37,72)(38,59)(39,70)(40,57)(41,68)(42,55)(43,66)(44,53)(45,64)(46,51)(47,62)(48,49) );
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)], [(1,65,32),(2,66,33),(3,67,34),(4,68,35),(5,69,36),(6,70,37),(7,71,38),(8,72,39),(9,49,40),(10,50,41),(11,51,42),(12,52,43),(13,53,44),(14,54,45),(15,55,46),(16,56,47),(17,57,48),(18,58,25),(19,59,26),(20,60,27),(21,61,28),(22,62,29),(23,63,30),(24,64,31)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,60),(26,71),(27,58),(28,69),(29,56),(30,67),(31,54),(32,65),(33,52),(34,63),(35,50),(36,61),(37,72),(38,59),(39,70),(40,57),(41,68),(42,55),(43,66),(44,53),(45,64),(46,51),(47,62),(48,49)]])
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