metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊C8⋊24D4, C3⋊5(C8⋊8D4), C4⋊C4.67D6, (C2×C6)⋊6SD16, C22⋊Q8⋊3S3, C4.174(S3×D4), (C2×Q8).53D6, C12.154(C2×D4), (C2×C12).267D4, C6.D8⋊39C2, C6.73(C2×SD16), (C22×C6).94D4, C6.101(C4○D8), C12⋊7D4.12C2, Q8⋊2Dic3⋊15C2, C6.97(C4⋊D4), C12.Q8⋊39C2, (C22×C4).358D6, C12.190(C4○D4), (C6×Q8).47C22, C4.63(D4⋊2S3), (C2×C12).367C23, C22⋊2(Q8⋊2S3), (C2×D12).99C22, C23.48(C3⋊D4), C2.20(Q8.13D6), C4⋊Dic3.146C22, C2.18(C23.14D6), (C22×C12).171C22, (C22×C3⋊C8)⋊5C2, (C3×C22⋊Q8)⋊3C2, (C2×C6).498(C2×D4), (C2×Q8⋊2S3)⋊10C2, (C2×C3⋊C8).250C22, C2.10(C2×Q8⋊2S3), (C2×C4).107(C3⋊D4), (C3×C4⋊C4).114C22, (C2×C4).467(C22×S3), C22.173(C2×C3⋊D4), SmallGroup(192,607)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊C8⋊24D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, bd=db, dcd=c-1 >
Subgroups: 368 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C2×C3⋊C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C8⋊8D4, C12.Q8, C6.D8, Q8⋊2Dic3, C22×C3⋊C8, C12⋊7D4, C2×Q8⋊2S3, C3×C22⋊Q8, C3⋊C8⋊24D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C4○D8, Q8⋊2S3, S3×D4, D4⋊2S3, C2×C3⋊D4, C8⋊8D4, C23.14D6, C2×Q8⋊2S3, Q8.13D6, C3⋊C8⋊24D4
►On 96 points
(1 70 57)(2 58 71)(3 72 59)(4 60 65)(5 66 61)(6 62 67)(7 68 63)(8 64 69)(9 92 37)(10 38 93)(11 94 39)(12 40 95)(13 96 33)(14 34 89)(15 90 35)(16 36 91)(17 81 74)(18 75 82)(19 83 76)(20 77 84)(21 85 78)(22 79 86)(23 87 80)(24 73 88)(25 44 49)(26 50 45)(27 46 51)(28 52 47)(29 48 53)(30 54 41)(31 42 55)(32 56 43)
(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 94 86 48)(2 89 87 43)(3 92 88 46)(4 95 81 41)(5 90 82 44)(6 93 83 47)(7 96 84 42)(8 91 85 45)(9 24 27 72)(10 19 28 67)(11 22 29 70)(12 17 30 65)(13 20 31 68)(14 23 32 71)(15 18 25 66)(16 21 26 69)(33 77 55 63)(34 80 56 58)(35 75 49 61)(36 78 50 64)(37 73 51 59)(38 76 52 62)(39 79 53 57)(40 74 54 60)
(1 94)(2 95)(3 96)(4 89)(5 90)(6 91)(7 92)(8 93)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 87)(42 88)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)
G:=sub<Sym(96)| (1,70,57)(2,58,71)(3,72,59)(4,60,65)(5,66,61)(6,62,67)(7,68,63)(8,64,69)(9,92,37)(10,38,93)(11,94,39)(12,40,95)(13,96,33)(14,34,89)(15,90,35)(16,36,91)(17,81,74)(18,75,82)(19,83,76)(20,77,84)(21,85,78)(22,79,86)(23,87,80)(24,73,88)(25,44,49)(26,50,45)(27,46,51)(28,52,47)(29,48,53)(30,54,41)(31,42,55)(32,56,43), (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), (1,94,86,48)(2,89,87,43)(3,92,88,46)(4,95,81,41)(5,90,82,44)(6,93,83,47)(7,96,84,42)(8,91,85,45)(9,24,27,72)(10,19,28,67)(11,22,29,70)(12,17,30,65)(13,20,31,68)(14,23,32,71)(15,18,25,66)(16,21,26,69)(33,77,55,63)(34,80,56,58)(35,75,49,61)(36,78,50,64)(37,73,51,59)(38,76,52,62)(39,79,53,57)(40,74,54,60), (1,94)(2,95)(3,96)(4,89)(5,90)(6,91)(7,92)(8,93)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)>;
G:=Group( (1,70,57)(2,58,71)(3,72,59)(4,60,65)(5,66,61)(6,62,67)(7,68,63)(8,64,69)(9,92,37)(10,38,93)(11,94,39)(12,40,95)(13,96,33)(14,34,89)(15,90,35)(16,36,91)(17,81,74)(18,75,82)(19,83,76)(20,77,84)(21,85,78)(22,79,86)(23,87,80)(24,73,88)(25,44,49)(26,50,45)(27,46,51)(28,52,47)(29,48,53)(30,54,41)(31,42,55)(32,56,43), (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), (1,94,86,48)(2,89,87,43)(3,92,88,46)(4,95,81,41)(5,90,82,44)(6,93,83,47)(7,96,84,42)(8,91,85,45)(9,24,27,72)(10,19,28,67)(11,22,29,70)(12,17,30,65)(13,20,31,68)(14,23,32,71)(15,18,25,66)(16,21,26,69)(33,77,55,63)(34,80,56,58)(35,75,49,61)(36,78,50,64)(37,73,51,59)(38,76,52,62)(39,79,53,57)(40,74,54,60), (1,94)(2,95)(3,96)(4,89)(5,90)(6,91)(7,92)(8,93)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,87)(42,88)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86) );
G=PermutationGroup([[(1,70,57),(2,58,71),(3,72,59),(4,60,65),(5,66,61),(6,62,67),(7,68,63),(8,64,69),(9,92,37),(10,38,93),(11,94,39),(12,40,95),(13,96,33),(14,34,89),(15,90,35),(16,36,91),(17,81,74),(18,75,82),(19,83,76),(20,77,84),(21,85,78),(22,79,86),(23,87,80),(24,73,88),(25,44,49),(26,50,45),(27,46,51),(28,52,47),(29,48,53),(30,54,41),(31,42,55),(32,56,43)], [(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)], [(1,94,86,48),(2,89,87,43),(3,92,88,46),(4,95,81,41),(5,90,82,44),(6,93,83,47),(7,96,84,42),(8,91,85,45),(9,24,27,72),(10,19,28,67),(11,22,29,70),(12,17,30,65),(13,20,31,68),(14,23,32,71),(15,18,25,66),(16,21,26,69),(33,77,55,63),(34,80,56,58),(35,75,49,61),(36,78,50,64),(37,73,51,59),(38,76,52,62),(39,79,53,57),(40,74,54,60)], [(1,94),(2,95),(3,96),(4,89),(5,90),(6,91),(7,92),(8,93),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,56),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(33,72),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,87),(42,88),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D6 | C4○D4 | SD16 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D4 | D4⋊2S3 | Q8⋊2S3 | Q8.13D6 |
| kernel | C3⋊C8⋊24D4 | C12.Q8 | C6.D8 | Q8⋊2Dic3 | C22×C3⋊C8 | C12⋊7D4 | C2×Q8⋊2S3 | C3×C22⋊Q8 | C22⋊Q8 | C3⋊C8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3⋊C8⋊24D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 30 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 69 | 0 | 0 |
| 0 | 0 | 31 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 6 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,43,0,0,0,0,30,60],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,59,31,0,0,0,0,69,14,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,14,6,0,0,0,0,4,59,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
C3⋊C8⋊24D4 in GAP, Magma, Sage, TeX
C_3\rtimes C_8\rtimes_{24}D_4 % in TeX
G:=Group("C3:C8:24D4"); // GroupNames label
G:=SmallGroup(192,607);
// by ID
G=gap.SmallGroup(192,607);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations