metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊33D4, D6⋊C8⋊3C2, C3⋊6(C8⋊9D4), D6⋊C4.3C4, C6.78(C4×D4), C8⋊15(C3⋊D4), Dic3⋊C8⋊3C2, (C22×C8)⋊15S3, C24⋊C4⋊23C2, (C2×C8).294D6, (C2×C6)⋊5M4(2), (C22×C24)⋊19C2, C6.19(C8○D4), C12.437(C2×D4), Dic3⋊C4.3C4, C23.40(C4×S3), C22⋊3(C8⋊S3), C2.19(C8○D12), C6.D4.9C4, (C22×C4).418D6, C6.13(C2×M4(2)), C4.136(C4○D12), C12.252(C4○D4), C12.55D4⋊26C2, (C2×C24).355C22, (C2×C12).861C23, (C22×C12).561C22, (C4×Dic3).187C22, (C2×C4).94(C4×S3), (C2×C3⋊D4).9C4, (C2×C8⋊S3)⋊23C2, C2.23(C4×C3⋊D4), C2.15(C2×C8⋊S3), (C4×C3⋊D4).15C2, C4.127(C2×C3⋊D4), C22.142(S3×C2×C4), (C2×C12).210(C2×C4), (C2×C3⋊C8).205C22, (S3×C2×C4).184C22, (C22×C6).96(C2×C4), (C22×S3).24(C2×C4), (C2×C4).803(C22×S3), (C2×C6).131(C22×C4), (C2×Dic3).32(C2×C4), SmallGroup(192,670)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊33D4
G = < a,b,c | a24=b4=c2=1, bab-1=cac=a5, cbc=b-1 >
Subgroups: 280 in 124 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8⋊9D4, Dic3⋊C8, C24⋊C4, D6⋊C8, C12.55D4, C2×C8⋊S3, C4×C3⋊D4, C22×C24, C24⋊33D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×M4(2), C8○D4, C8⋊S3, S3×C2×C4, C4○D12, C2×C3⋊D4, C8⋊9D4, C2×C8⋊S3, C8○D12, C4×C3⋊D4, C24⋊33D4
►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 36 95 66)(2 41 96 71)(3 46 73 52)(4 27 74 57)(5 32 75 62)(6 37 76 67)(7 42 77 72)(8 47 78 53)(9 28 79 58)(10 33 80 63)(11 38 81 68)(12 43 82 49)(13 48 83 54)(14 29 84 59)(15 34 85 64)(16 39 86 69)(17 44 87 50)(18 25 88 55)(19 30 89 60)(20 35 90 65)(21 40 91 70)(22 45 92 51)(23 26 93 56)(24 31 94 61)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 59)(26 64)(27 69)(28 50)(29 55)(30 60)(31 65)(32 70)(33 51)(34 56)(35 61)(36 66)(37 71)(38 52)(39 57)(40 62)(41 67)(42 72)(43 53)(44 58)(45 63)(46 68)(47 49)(48 54)(73 81)(74 86)(75 91)(76 96)(78 82)(79 87)(80 92)(84 88)(85 93)(90 94)
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), (1,36,95,66)(2,41,96,71)(3,46,73,52)(4,27,74,57)(5,32,75,62)(6,37,76,67)(7,42,77,72)(8,47,78,53)(9,28,79,58)(10,33,80,63)(11,38,81,68)(12,43,82,49)(13,48,83,54)(14,29,84,59)(15,34,85,64)(16,39,86,69)(17,44,87,50)(18,25,88,55)(19,30,89,60)(20,35,90,65)(21,40,91,70)(22,45,92,51)(23,26,93,56)(24,31,94,61), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,59)(26,64)(27,69)(28,50)(29,55)(30,60)(31,65)(32,70)(33,51)(34,56)(35,61)(36,66)(37,71)(38,52)(39,57)(40,62)(41,67)(42,72)(43,53)(44,58)(45,63)(46,68)(47,49)(48,54)(73,81)(74,86)(75,91)(76,96)(78,82)(79,87)(80,92)(84,88)(85,93)(90,94)>;
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), (1,36,95,66)(2,41,96,71)(3,46,73,52)(4,27,74,57)(5,32,75,62)(6,37,76,67)(7,42,77,72)(8,47,78,53)(9,28,79,58)(10,33,80,63)(11,38,81,68)(12,43,82,49)(13,48,83,54)(14,29,84,59)(15,34,85,64)(16,39,86,69)(17,44,87,50)(18,25,88,55)(19,30,89,60)(20,35,90,65)(21,40,91,70)(22,45,92,51)(23,26,93,56)(24,31,94,61), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,59)(26,64)(27,69)(28,50)(29,55)(30,60)(31,65)(32,70)(33,51)(34,56)(35,61)(36,66)(37,71)(38,52)(39,57)(40,62)(41,67)(42,72)(43,53)(44,58)(45,63)(46,68)(47,49)(48,54)(73,81)(74,86)(75,91)(76,96)(78,82)(79,87)(80,92)(84,88)(85,93)(90,94) );
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)], [(1,36,95,66),(2,41,96,71),(3,46,73,52),(4,27,74,57),(5,32,75,62),(6,37,76,67),(7,42,77,72),(8,47,78,53),(9,28,79,58),(10,33,80,63),(11,38,81,68),(12,43,82,49),(13,48,83,54),(14,29,84,59),(15,34,85,64),(16,39,86,69),(17,44,87,50),(18,25,88,55),(19,30,89,60),(20,35,90,65),(21,40,91,70),(22,45,92,51),(23,26,93,56),(24,31,94,61)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,59),(26,64),(27,69),(28,50),(29,55),(30,60),(31,65),(32,70),(33,51),(34,56),(35,61),(36,66),(37,71),(38,52),(39,57),(40,62),(41,67),(42,72),(43,53),(44,58),(45,63),(46,68),(47,49),(48,54),(73,81),(74,86),(75,91),(76,96),(78,82),(79,87),(80,92),(84,88),(85,93),(90,94)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | M4(2) | C3⋊D4 | C4×S3 | C4×S3 | C8○D4 | C4○D12 | C8⋊S3 | C8○D12 |
| kernel | C24⋊33D4 | Dic3⋊C8 | C24⋊C4 | D6⋊C8 | C12.55D4 | C2×C8⋊S3 | C4×C3⋊D4 | C22×C24 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22×C8 | C24 | C2×C8 | C22×C4 | C12 | C2×C6 | C8 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C24⋊33D4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 6 | 70 |
| 0 | 0 | 3 | 3 |
| 1 | 2 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 60 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,6,3,0,0,70,3],[1,72,0,0,2,72,0,0,0,0,13,43,0,0,30,60],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
C24⋊33D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{33}D_4 % in TeX
G:=Group("C24:33D4"); // GroupNames label
G:=SmallGroup(192,670);
// by ID
G=gap.SmallGroup(192,670);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations