metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊D4⋊C8, C3⋊2(C8×D4), C3⋊C8⋊29D4, D6⋊2(C2×C8), D6⋊C8⋊21C2, D6⋊C4.9C4, C22⋊2(S3×C8), C6.26(C4×D4), C22⋊C8⋊16S3, Dic3⋊1(C2×C8), (C2×C8).194D6, C4.194(S3×D4), C6.7(C22×C8), Dic3⋊C8⋊23C2, (C8×Dic3)⋊15C2, C6.24(C8○D4), C12.353(C2×D4), Dic3⋊C4.9C4, C23.28(C4×S3), C2.2(D12.C4), C6.D4.4C4, (C22×C4).319D6, C12.297(C4○D4), (C2×C12).820C23, (C2×C24).213C22, C4.123(D4⋊2S3), C2.3(Dic3⋊4D4), (C22×C12).337C22, (C4×Dic3).271C22, C2.9(S3×C2×C8), (S3×C2×C8)⋊13C2, (C2×C6)⋊2(C2×C8), (C22×C3⋊C8)⋊16C2, (C2×C4).63(C4×S3), (C2×C3⋊D4).3C4, C22.44(S3×C2×C4), (C3×C22⋊C8)⋊19C2, (C4×C3⋊D4).13C2, (C2×C12).153(C2×C4), (C2×C3⋊C8).300C22, (S3×C2×C4).272C22, (C2×C6).75(C22×C4), (C22×C6).38(C2×C4), (C22×S3).35(C2×C4), (C2×C4).762(C22×S3), (C2×Dic3).49(C2×C4), SmallGroup(192,284)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D4⋊C8
G = < a,b,c,d | a3=b4=c2=d8=1, bab-1=cac=a-1, ad=da, cbc=dbd-1=b-1, cd=dc >
Subgroups: 288 in 134 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, S3×C8, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8×D4, C8×Dic3, Dic3⋊C8, D6⋊C8, C3×C22⋊C8, S3×C2×C8, C22×C3⋊C8, C4×C3⋊D4, C3⋊D4⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C2×C8, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C22×C8, C8○D4, S3×C8, S3×C2×C4, S3×D4, D4⋊2S3, C8×D4, Dic3⋊4D4, S3×C2×C8, D12.C4, C3⋊D4⋊C8
►On 96 points
(1 73 43)(2 74 44)(3 75 45)(4 76 46)(5 77 47)(6 78 48)(7 79 41)(8 80 42)(9 39 72)(10 40 65)(11 33 66)(12 34 67)(13 35 68)(14 36 69)(15 37 70)(16 38 71)(17 60 85)(18 61 86)(19 62 87)(20 63 88)(21 64 81)(22 57 82)(23 58 83)(24 59 84)(25 52 92)(26 53 93)(27 54 94)(28 55 95)(29 56 96)(30 49 89)(31 50 90)(32 51 91)
(1 72 23 27)(2 28 24 65)(3 66 17 29)(4 30 18 67)(5 68 19 31)(6 32 20 69)(7 70 21 25)(8 26 22 71)(9 83 54 43)(10 44 55 84)(11 85 56 45)(12 46 49 86)(13 87 50 47)(14 48 51 88)(15 81 52 41)(16 42 53 82)(33 60 96 75)(34 76 89 61)(35 62 90 77)(36 78 91 63)(37 64 92 79)(38 80 93 57)(39 58 94 73)(40 74 95 59)
(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 81)
(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)
G:=sub<Sym(96)| (1,73,43)(2,74,44)(3,75,45)(4,76,46)(5,77,47)(6,78,48)(7,79,41)(8,80,42)(9,39,72)(10,40,65)(11,33,66)(12,34,67)(13,35,68)(14,36,69)(15,37,70)(16,38,71)(17,60,85)(18,61,86)(19,62,87)(20,63,88)(21,64,81)(22,57,82)(23,58,83)(24,59,84)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,49,89)(31,50,90)(32,51,91), (1,72,23,27)(2,28,24,65)(3,66,17,29)(4,30,18,67)(5,68,19,31)(6,32,20,69)(7,70,21,25)(8,26,22,71)(9,83,54,43)(10,44,55,84)(11,85,56,45)(12,46,49,86)(13,87,50,47)(14,48,51,88)(15,81,52,41)(16,42,53,82)(33,60,96,75)(34,76,89,61)(35,62,90,77)(36,78,91,63)(37,64,92,79)(38,80,93,57)(39,58,94,73)(40,74,95,59), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81), (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)>;
G:=Group( (1,73,43)(2,74,44)(3,75,45)(4,76,46)(5,77,47)(6,78,48)(7,79,41)(8,80,42)(9,39,72)(10,40,65)(11,33,66)(12,34,67)(13,35,68)(14,36,69)(15,37,70)(16,38,71)(17,60,85)(18,61,86)(19,62,87)(20,63,88)(21,64,81)(22,57,82)(23,58,83)(24,59,84)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,49,89)(31,50,90)(32,51,91), (1,72,23,27)(2,28,24,65)(3,66,17,29)(4,30,18,67)(5,68,19,31)(6,32,20,69)(7,70,21,25)(8,26,22,71)(9,83,54,43)(10,44,55,84)(11,85,56,45)(12,46,49,86)(13,87,50,47)(14,48,51,88)(15,81,52,41)(16,42,53,82)(33,60,96,75)(34,76,89,61)(35,62,90,77)(36,78,91,63)(37,64,92,79)(38,80,93,57)(39,58,94,73)(40,74,95,59), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,81), (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) );
G=PermutationGroup([[(1,73,43),(2,74,44),(3,75,45),(4,76,46),(5,77,47),(6,78,48),(7,79,41),(8,80,42),(9,39,72),(10,40,65),(11,33,66),(12,34,67),(13,35,68),(14,36,69),(15,37,70),(16,38,71),(17,60,85),(18,61,86),(19,62,87),(20,63,88),(21,64,81),(22,57,82),(23,58,83),(24,59,84),(25,52,92),(26,53,93),(27,54,94),(28,55,95),(29,56,96),(30,49,89),(31,50,90),(32,51,91)], [(1,72,23,27),(2,28,24,65),(3,66,17,29),(4,30,18,67),(5,68,19,31),(6,32,20,69),(7,70,21,25),(8,26,22,71),(9,83,54,43),(10,44,55,84),(11,85,56,45),(12,46,49,86),(13,87,50,47),(14,48,51,88),(15,81,52,41),(16,42,53,82),(33,60,96,75),(34,76,89,61),(35,62,90,77),(36,78,91,63),(37,64,92,79),(38,80,93,57),(39,58,94,73),(40,74,95,59)], [(9,94),(10,95),(11,96),(12,89),(13,90),(14,91),(15,92),(16,93),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,79),(42,80),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,81)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | C8○D4 | S3×C8 | S3×D4 | D4⋊2S3 | D12.C4 |
| kernel | C3⋊D4⋊C8 | C8×Dic3 | Dic3⋊C8 | D6⋊C8 | C3×C22⋊C8 | S3×C2×C8 | C22×C3⋊C8 | C4×C3⋊D4 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C3⋊D4 | C22⋊C8 | C3⋊C8 | C2×C8 | C22×C4 | C12 | C2×C4 | C23 | C6 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C3⋊D4⋊C8 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0],[10,0,0,0,0,0,51,0,0,0,0,0,22,0,0,0,0,0,72,0,0,0,0,0,72] >;
C3⋊D4⋊C8 in GAP, Magma, Sage, TeX
C_3\rtimes D_4\rtimes C_8
% in TeX
G:=Group("C3:D4:C8"); // GroupNames label
G:=SmallGroup(192,284);
// by ID
G=gap.SmallGroup(192,284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^2=d^8=1,b*a*b^-1=c*a*c=a^-1,a*d=d*a,c*b*c=d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations