direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D8⋊C4, D8⋊3C12, C8⋊3(C2×C12), (C3×D8)⋊9C4, (C4×D4)⋊2C6, C4.Q8⋊3C6, C8⋊C4⋊2C6, C24⋊18(C2×C4), D4⋊2(C2×C12), (C2×D8).6C6, (D4×C12)⋊31C2, (C6×D8).13C2, C2.17(D4×C12), C6.119(C4×D4), D4⋊C4⋊16C6, (C2×C12).458D4, C42.10(C2×C6), C22.56(C6×D4), C4.14(C22×C12), C12.261(C4○D4), C6.130(C8⋊C22), (C2×C12).909C23, C12.159(C22×C4), (C2×C24).331C22, (C4×C12).251C22, (C6×D4).293C22, (C3×C8⋊C4)⋊7C2, C4.6(C3×C4○D4), (C3×D4)⋊15(C2×C4), C4⋊C4.50(C2×C6), (C2×C8).20(C2×C6), (C3×C4.Q8)⋊12C2, C2.5(C3×C8⋊C22), (C2×D4).51(C2×C6), (C2×C6).632(C2×D4), (C2×C4).104(C3×D4), (C3×D4⋊C4)⋊39C2, (C2×C4).84(C22×C6), (C3×C4⋊C4).371C22, SmallGroup(192,875)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8⋊C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >
Subgroups: 250 in 132 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C22×C12, C6×D4, D8⋊C4, C3×C8⋊C4, C3×D4⋊C4, C3×C4.Q8, D4×C12, C6×D8, C3×D8⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C8⋊C22, C22×C12, C6×D4, C3×C4○D4, D8⋊C4, D4×C12, C3×C8⋊C22, C3×D8⋊C4
►On 96 points
Generators in S96
(1 79 95)(2 80 96)(3 73 89)(4 74 90)(5 75 91)(6 76 92)(7 77 93)(8 78 94)(9 68 84)(10 69 85)(11 70 86)(12 71 87)(13 72 88)(14 65 81)(15 66 82)(16 67 83)(17 28 55)(18 29 56)(19 30 49)(20 31 50)(21 32 51)(22 25 52)(23 26 53)(24 27 54)(33 42 57)(34 43 58)(35 44 59)(36 45 60)(37 46 61)(38 47 62)(39 48 63)(40 41 64)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 22)(18 21)(19 20)(23 24)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)
(1 71 27 43)(2 68 28 48)(3 65 29 45)(4 70 30 42)(5 67 31 47)(6 72 32 44)(7 69 25 41)(8 66 26 46)(9 17 39 96)(10 22 40 93)(11 19 33 90)(12 24 34 95)(13 21 35 92)(14 18 36 89)(15 23 37 94)(16 20 38 91)(49 57 74 86)(50 62 75 83)(51 59 76 88)(52 64 77 85)(53 61 78 82)(54 58 79 87)(55 63 80 84)(56 60 73 81)
G:=sub<Sym(96)| (1,79,95)(2,80,96)(3,73,89)(4,74,90)(5,75,91)(6,76,92)(7,77,93)(8,78,94)(9,68,84)(10,69,85)(11,70,86)(12,71,87)(13,72,88)(14,65,81)(15,66,82)(16,67,83)(17,28,55)(18,29,56)(19,30,49)(20,31,50)(21,32,51)(22,25,52)(23,26,53)(24,27,54)(33,42,57)(34,43,58)(35,44,59)(36,45,60)(37,46,61)(38,47,62)(39,48,63)(40,41,64), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95), (1,71,27,43)(2,68,28,48)(3,65,29,45)(4,70,30,42)(5,67,31,47)(6,72,32,44)(7,69,25,41)(8,66,26,46)(9,17,39,96)(10,22,40,93)(11,19,33,90)(12,24,34,95)(13,21,35,92)(14,18,36,89)(15,23,37,94)(16,20,38,91)(49,57,74,86)(50,62,75,83)(51,59,76,88)(52,64,77,85)(53,61,78,82)(54,58,79,87)(55,63,80,84)(56,60,73,81)>;
G:=Group( (1,79,95)(2,80,96)(3,73,89)(4,74,90)(5,75,91)(6,76,92)(7,77,93)(8,78,94)(9,68,84)(10,69,85)(11,70,86)(12,71,87)(13,72,88)(14,65,81)(15,66,82)(16,67,83)(17,28,55)(18,29,56)(19,30,49)(20,31,50)(21,32,51)(22,25,52)(23,26,53)(24,27,54)(33,42,57)(34,43,58)(35,44,59)(36,45,60)(37,46,61)(38,47,62)(39,48,63)(40,41,64), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)(86,87)(89,92)(90,91)(93,96)(94,95), (1,71,27,43)(2,68,28,48)(3,65,29,45)(4,70,30,42)(5,67,31,47)(6,72,32,44)(7,69,25,41)(8,66,26,46)(9,17,39,96)(10,22,40,93)(11,19,33,90)(12,24,34,95)(13,21,35,92)(14,18,36,89)(15,23,37,94)(16,20,38,91)(49,57,74,86)(50,62,75,83)(51,59,76,88)(52,64,77,85)(53,61,78,82)(54,58,79,87)(55,63,80,84)(56,60,73,81) );
G=PermutationGroup([[(1,79,95),(2,80,96),(3,73,89),(4,74,90),(5,75,91),(6,76,92),(7,77,93),(8,78,94),(9,68,84),(10,69,85),(11,70,86),(12,71,87),(13,72,88),(14,65,81),(15,66,82),(16,67,83),(17,28,55),(18,29,56),(19,30,49),(20,31,50),(21,32,51),(22,25,52),(23,26,53),(24,27,54),(33,42,57),(34,43,58),(35,44,59),(36,45,60),(37,46,61),(38,47,62),(39,48,63),(40,41,64)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,22),(18,21),(19,20),(23,24),(25,28),(26,27),(29,32),(30,31),(33,34),(35,40),(36,39),(37,38),(41,44),(42,43),(45,48),(46,47),(49,50),(51,56),(52,55),(53,54),(57,58),(59,64),(60,63),(61,62),(65,68),(66,67),(69,72),(70,71),(73,76),(74,75),(77,80),(78,79),(81,84),(82,83),(85,88),(86,87),(89,92),(90,91),(93,96),(94,95)], [(1,71,27,43),(2,68,28,48),(3,65,29,45),(4,70,30,42),(5,67,31,47),(6,72,32,44),(7,69,25,41),(8,66,26,46),(9,17,39,96),(10,22,40,93),(11,19,33,90),(12,24,34,95),(13,21,35,92),(14,18,36,89),(15,23,37,94),(16,20,38,91),(49,57,74,86),(50,62,75,83),(51,59,76,88),(52,64,77,85),(53,61,78,82),(54,58,79,87),(55,63,80,84),(56,60,73,81)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6N | 8A | 8B | 8C | 8D | 12A | ··· | 12L | 12M | ··· | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×D8⋊C4 | C3×C8⋊C4 | C3×D4⋊C4 | C3×C4.Q8 | D4×C12 | C6×D8 | D8⋊C4 | C3×D8 | C8⋊C4 | D4⋊C4 | C4.Q8 | C4×D4 | C2×D8 | D8 | C2×C12 | C12 | C2×C4 | C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×D8⋊C4 ►in GL8(𝔽73)
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 33 | 0 | 24 |
| 0 | 0 | 0 | 0 | 20 | 33 | 61 | 24 |
| 0 | 0 | 0 | 0 | 0 | 49 | 0 | 40 |
| 0 | 0 | 0 | 0 | 12 | 49 | 53 | 40 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 33 | 0 | 24 |
| 0 | 0 | 0 | 0 | 53 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 49 | 0 | 40 |
| 0 | 0 | 0 | 0 | 61 | 0 | 20 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,17,54,0,0,0,0,0,0,46,56,0,0,0,0,0,0,0,0,0,20,0,12,0,0,0,0,33,33,49,49,0,0,0,0,0,61,0,53,0,0,0,0,24,24,40,40],[1,0,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,56,19,0,0,0,0,0,0,54,17,0,0,0,0,0,0,0,0,0,53,0,61,0,0,0,0,33,0,49,0,0,0,0,0,0,12,0,20,0,0,0,0,24,0,40,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C3×D8⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_8\rtimes C_4
% in TeX
G:=Group("C3xD8:C4"); // GroupNames label
G:=SmallGroup(192,875);
// by ID
G=gap.SmallGroup(192,875);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,772,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations