direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊3D4, C24⋊18D4, C8⋊3(C3×D4), (C2×D8)⋊9C6, C8⋊C4⋊4C6, C4.5(C6×D4), (C6×D8)⋊23C2, C4⋊1D4⋊6C6, (C2×SD16)⋊3C6, C4.4D4⋊5C6, (C6×SD16)⋊14C2, C12.312(C2×D4), (C2×C12).343D4, C42.29(C2×C6), C6.46(C4⋊1D4), C22.117(C6×D4), C6.147(C8⋊C22), (C2×C24).203C22, (C2×C12).952C23, (C4×C12).271C22, (C6×D4).205C22, (C6×Q8).179C22, (C3×C8⋊C4)⋊9C2, (C2×C8).27(C2×C6), (C2×C4).44(C3×D4), C2.9(C3×C4⋊1D4), (C3×C4⋊1D4)⋊14C2, (C2×D4).28(C2×C6), (C2×C6).673(C2×D4), C2.22(C3×C8⋊C22), (C2×Q8).24(C2×C6), (C3×C4.4D4)⋊25C2, (C2×C4).127(C22×C6), SmallGroup(192,929)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊3D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 322 in 144 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×D4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C4×C12, C3×C22⋊C4, C2×C24, C3×D8, C3×SD16, C6×D4, C6×D4, C6×D4, C6×Q8, C8⋊3D4, C3×C8⋊C4, C3×C4.4D4, C3×C4⋊1D4, C6×D8, C6×SD16, C3×C8⋊3D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C4⋊1D4, C8⋊C22, C6×D4, C8⋊3D4, C3×C4⋊1D4, C3×C8⋊C22, C3×C8⋊3D4
►On 96 points
Generators in S96
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 32 62)(10 25 63)(11 26 64)(12 27 57)(13 28 58)(14 29 59)(15 30 60)(16 31 61)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 55 89)(42 56 90)(43 49 91)(44 50 92)(45 51 93)(46 52 94)(47 53 95)(48 54 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 47 79 25)(2 44 80 30)(3 41 73 27)(4 46 74 32)(5 43 75 29)(6 48 76 26)(7 45 77 31)(8 42 78 28)(9 20 94 34)(10 17 95 39)(11 22 96 36)(12 19 89 33)(13 24 90 38)(14 21 91 35)(15 18 92 40)(16 23 93 37)(49 83 59 69)(50 88 60 66)(51 85 61 71)(52 82 62 68)(53 87 63 65)(54 84 64 70)(55 81 57 67)(56 86 58 72)
(1 74)(2 73)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 10)(11 16)(12 15)(13 14)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 32)(26 31)(27 30)(28 29)(41 44)(42 43)(45 48)(46 47)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 82)(66 81)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(89 92)(90 91)(93 96)(94 95)
G:=sub<Sym(96)| (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,62)(10,25,63)(11,26,64)(12,27,57)(13,28,58)(14,29,59)(15,30,60)(16,31,61)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (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), (1,47,79,25)(2,44,80,30)(3,41,73,27)(4,46,74,32)(5,43,75,29)(6,48,76,26)(7,45,77,31)(8,42,78,28)(9,20,94,34)(10,17,95,39)(11,22,96,36)(12,19,89,33)(13,24,90,38)(14,21,91,35)(15,18,92,40)(16,23,93,37)(49,83,59,69)(50,88,60,66)(51,85,61,71)(52,82,62,68)(53,87,63,65)(54,84,64,70)(55,81,57,67)(56,86,58,72), (1,74)(2,73)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,10)(11,16)(12,15)(13,14)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,32)(26,31)(27,30)(28,29)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,64)(62,63)(65,82)(66,81)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(89,92)(90,91)(93,96)(94,95)>;
G:=Group( (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,62)(10,25,63)(11,26,64)(12,27,57)(13,28,58)(14,29,59)(15,30,60)(16,31,61)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (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), (1,47,79,25)(2,44,80,30)(3,41,73,27)(4,46,74,32)(5,43,75,29)(6,48,76,26)(7,45,77,31)(8,42,78,28)(9,20,94,34)(10,17,95,39)(11,22,96,36)(12,19,89,33)(13,24,90,38)(14,21,91,35)(15,18,92,40)(16,23,93,37)(49,83,59,69)(50,88,60,66)(51,85,61,71)(52,82,62,68)(53,87,63,65)(54,84,64,70)(55,81,57,67)(56,86,58,72), (1,74)(2,73)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,10)(11,16)(12,15)(13,14)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,32)(26,31)(27,30)(28,29)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,64)(62,63)(65,82)(66,81)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(89,92)(90,91)(93,96)(94,95) );
G=PermutationGroup([[(1,65,17),(2,66,18),(3,67,19),(4,68,20),(5,69,21),(6,70,22),(7,71,23),(8,72,24),(9,32,62),(10,25,63),(11,26,64),(12,27,57),(13,28,58),(14,29,59),(15,30,60),(16,31,61),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,55,89),(42,56,90),(43,49,91),(44,50,92),(45,51,93),(46,52,94),(47,53,95),(48,54,96)], [(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)], [(1,47,79,25),(2,44,80,30),(3,41,73,27),(4,46,74,32),(5,43,75,29),(6,48,76,26),(7,45,77,31),(8,42,78,28),(9,20,94,34),(10,17,95,39),(11,22,96,36),(12,19,89,33),(13,24,90,38),(14,21,91,35),(15,18,92,40),(16,23,93,37),(49,83,59,69),(50,88,60,66),(51,85,61,71),(52,82,62,68),(53,87,63,65),(54,84,64,70),(55,81,57,67),(56,86,58,72)], [(1,74),(2,73),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,10),(11,16),(12,15),(13,14),(17,34),(18,33),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,32),(26,31),(27,30),(28,29),(41,44),(42,43),(45,48),(46,47),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,64),(62,63),(65,82),(66,81),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(89,92),(90,91),(93,96),(94,95)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6F | 6G | ··· | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | ··· | 1 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C8⋊3D4 | C3×C8⋊C4 | C3×C4.4D4 | C3×C4⋊1D4 | C6×D8 | C6×SD16 | C8⋊3D4 | C8⋊C4 | C4.4D4 | C4⋊1D4 | C2×D8 | C2×SD16 | C24 | C2×C12 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 2 | 4 |
Matrix representation of C3×C8⋊3D4 ►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 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 2 | 19 | 54 |
| 0 | 0 | 0 | 0 | 71 | 2 | 0 | 54 |
| 0 | 0 | 0 | 0 | 2 | 17 | 15 | 58 |
| 0 | 0 | 0 | 0 | 17 | 0 | 71 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 1 | 71 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 2 | 19 | 54 |
| 0 | 0 | 0 | 0 | 2 | 71 | 0 | 19 |
| 0 | 0 | 0 | 0 | 2 | 17 | 15 | 58 |
| 0 | 0 | 0 | 0 | 2 | 15 | 71 | 4 |
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],[0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,56,71,2,17,0,0,0,0,2,2,17,0,0,0,0,0,19,0,15,71,0,0,0,0,54,54,58,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,72,72,72,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,71,0,72],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,56,2,2,2,0,0,0,0,2,71,17,15,0,0,0,0,19,0,15,71,0,0,0,0,54,19,58,4] >;
C3×C8⋊3D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_3D_4
% in TeX
G:=Group("C3xC8:3D4"); // GroupNames label
G:=SmallGroup(192,929);
// by ID
G=gap.SmallGroup(192,929);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,1059,268,4204,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations