metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24:2D4, C23.22D12, C3:6(C8:D4), C8:1(C3:D4), (C2xC8).77D6, C24:1C4:18C2, (C2xC4).51D12, C12.420(C2xD4), (C2xC12).297D4, C2.D24:41C2, (C2xM4(2)):1S3, (C6xM4(2)):1C2, C12:7D4.17C2, C6.73(C4:D4), C2.22(C8:D6), C6.22(C8:C22), (C2xC24).63C22, C2.Dic12:41C2, (C22xC4).158D6, (C22xC6).103D4, C12.230(C4oD4), C4.114(C4oD12), C12.48D4:41C2, C2.21(C12:7D4), (C2xC12).775C23, C2.22(C8.D6), (C2xD12).20C22, C22.134(C2xD12), C6.22(C8.C22), C4:Dic3.26C22, (C2xDic6).19C22, (C22xC12).304C22, (C2xC24:C2):2C2, (C2xC6).165(C2xD4), C4.113(C2xC3:D4), (C2xC4).724(C22xS3), SmallGroup(192,693)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:2D4
G = < a,b,c | a24=b4=c2=1, bab-1=a-1, cac=a11, cbc=b-1 >
Subgroups: 392 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, M4(2), SD16, C22xC4, C2xD4, C2xQ8, C24, C24, Dic6, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xC6, D4:C4, Q8:C4, C2.D8, C4:D4, C22:Q8, C2xM4(2), C2xSD16, C24:C2, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C2xC24, C3xM4(2), C2xDic6, C2xD12, C2xC3:D4, C22xC12, C8:D4, C2.Dic12, C24:1C4, C2.D24, C2xC24:C2, C12.48D4, C12:7D4, C6xM4(2), C24:2D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, C8:C22, C8.C22, C2xD12, C4oD12, C2xC3:D4, C8:D4, C8:D6, C8.D6, C12:7D4, C24:2D4
►On 96 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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 53 88 45)(2 52 89 44)(3 51 90 43)(4 50 91 42)(5 49 92 41)(6 72 93 40)(7 71 94 39)(8 70 95 38)(9 69 96 37)(10 68 73 36)(11 67 74 35)(12 66 75 34)(13 65 76 33)(14 64 77 32)(15 63 78 31)(16 62 79 30)(17 61 80 29)(18 60 81 28)(19 59 82 27)(20 58 83 26)(21 57 84 25)(22 56 85 48)(23 55 86 47)(24 54 87 46)
(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)(73 91)(74 78)(75 89)(77 87)(79 85)(80 96)(81 83)(82 94)(84 92)(86 90)(93 95)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,53,88,45)(2,52,89,44)(3,51,90,43)(4,50,91,42)(5,49,92,41)(6,72,93,40)(7,71,94,39)(8,70,95,38)(9,69,96,37)(10,68,73,36)(11,67,74,35)(12,66,75,34)(13,65,76,33)(14,64,77,32)(15,63,78,31)(16,62,79,30)(17,61,80,29)(18,60,81,28)(19,59,82,27)(20,58,83,26)(21,57,84,25)(22,56,85,48)(23,55,86,47)(24,54,87,46), (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)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,53,88,45)(2,52,89,44)(3,51,90,43)(4,50,91,42)(5,49,92,41)(6,72,93,40)(7,71,94,39)(8,70,95,38)(9,69,96,37)(10,68,73,36)(11,67,74,35)(12,66,75,34)(13,65,76,33)(14,64,77,32)(15,63,78,31)(16,62,79,30)(17,61,80,29)(18,60,81,28)(19,59,82,27)(20,58,83,26)(21,57,84,25)(22,56,85,48)(23,55,86,47)(24,54,87,46), (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)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95) );
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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,53,88,45),(2,52,89,44),(3,51,90,43),(4,50,91,42),(5,49,92,41),(6,72,93,40),(7,71,94,39),(8,70,95,38),(9,69,96,37),(10,68,73,36),(11,67,74,35),(12,66,75,34),(13,65,76,33),(14,64,77,32),(15,63,78,31),(16,62,79,30),(17,61,80,29),(18,60,81,28),(19,59,82,27),(20,58,83,26),(21,57,84,25),(22,56,85,48),(23,55,86,47),(24,54,87,46)], [(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),(73,91),(74,78),(75,89),(77,87),(79,85),(80,96),(81,83),(82,94),(84,92),(86,90),(93,95)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 24 | 2 | 2 | 2 | 4 | 24 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4oD4 | C3:D4 | D12 | D12 | C4oD12 | C8:C22 | C8.C22 | C8:D6 | C8.D6 |
| kernel | C24:2D4 | C2.Dic12 | C24:1C4 | C2.D24 | C2xC24:C2 | C12.48D4 | C12:7D4 | C6xM4(2) | C2xM4(2) | C24 | C2xC12 | C22xC6 | C2xC8 | C22xC4 | C12 | C8 | C2xC4 | C23 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C24:2D4 ►in GL8(F73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 3 | 6 | 6 |
| 0 | 0 | 0 | 0 | 41 | 53 | 18 | 0 |
| 0 | 0 | 0 | 0 | 52 | 52 | 55 | 35 |
| 0 | 0 | 0 | 0 | 62 | 11 | 26 | 55 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 28 | 17 | 3 | 3 |
| 0 | 0 | 0 | 0 | 33 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 70 | 25 | 71 | 56 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 56 | 60 | 0 | 1 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,56,41,52,62,0,0,0,0,3,53,52,11,0,0,0,0,6,18,55,26,0,0,0,0,6,0,35,55],[1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,58,28,33,70,0,0,0,0,0,17,0,25,0,0,0,0,2,3,15,71,0,0,0,0,0,3,0,56],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,15,56,0,0,0,0,0,72,0,60,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1] >;
C24:2D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_2D_4 % in TeX
G:=Group("C24:2D4"); // GroupNames label
G:=SmallGroup(192,693);
// by ID
G=gap.SmallGroup(192,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations