direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.7D4, C22⋊C8⋊6C6, (C2×Q16)⋊2C6, D4.7(C3×D4), C4.26(C6×D4), C22⋊Q8⋊2C6, D4⋊C4⋊5C6, (C6×Q16)⋊16C2, (C3×D4).41D4, Q8.12(C3×D4), (C3×Q8).41D4, Q8⋊C4⋊10C6, C6.99C22≀C2, (C6×SD16)⋊27C2, (C2×SD16)⋊10C6, (C2×C12).460D4, C12.387(C2×D4), C23.14(C3×D4), (C22×C6).32D4, C22.82(C6×D4), C6.120(C4○D8), (C2×C24).183C22, (C2×C12).917C23, (C6×D4).296C22, (C6×Q8).261C22, C6.133(C8.C22), (C22×C12).424C22, C4⋊C4.4(C2×C6), C2.7(C3×C4○D8), (C2×C8).35(C2×C6), (C3×C22⋊C8)⋊16C2, (C6×C4○D4).22C2, (C2×C4○D4).14C6, (C2×D4).54(C2×C6), (C2×C4).106(C3×D4), (C2×C6).638(C2×D4), (C3×C22⋊Q8)⋊29C2, (C2×Q8).58(C2×C6), (C3×D4⋊C4)⋊16C2, C2.8(C3×C8.C22), (C3×Q8⋊C4)⋊32C2, C2.13(C3×C22≀C2), (C22×C4).47(C2×C6), (C2×C4).92(C22×C6), (C3×C4⋊C4).226C22, SmallGroup(192,885)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.7D4
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >
Subgroups: 274 in 152 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C3×Q16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, D4.7D4, C3×C22⋊C8, C3×D4⋊C4, C3×Q8⋊C4, C3×C22⋊Q8, C6×SD16, C6×Q16, C6×C4○D4, C3×D4.7D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C4○D8, C8.C22, C6×D4, D4.7D4, C3×C22≀C2, C3×C4○D8, C3×C8.C22, C3×D4.7D4
►On 96 points
Generators in S96
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 91 83)(6 92 84)(7 89 81)(8 90 82)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(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 35)(2 34)(3 33)(4 36)(5 70)(6 69)(7 72)(8 71)(9 29)(10 32)(11 31)(12 30)(13 42)(14 41)(15 44)(16 43)(17 37)(18 40)(19 39)(20 38)(21 50)(22 49)(23 52)(24 51)(25 48)(26 47)(27 46)(28 45)(53 84)(54 83)(55 82)(56 81)(57 77)(58 80)(59 79)(60 78)(61 92)(62 91)(63 90)(64 89)(65 85)(66 88)(67 87)(68 86)(73 94)(74 93)(75 96)(76 95)
(1 59 11 55)(2 58 12 54)(3 57 9 53)(4 60 10 56)(5 49 96 45)(6 52 93 48)(7 51 94 47)(8 50 95 46)(13 66 20 62)(14 65 17 61)(15 68 18 64)(16 67 19 63)(21 75 27 70)(22 74 28 69)(23 73 25 72)(24 76 26 71)(29 83 33 80)(30 82 34 79)(31 81 35 78)(32 84 36 77)(37 91 41 88)(38 90 42 87)(39 89 43 86)(40 92 44 85)
(1 54 3 56)(2 53 4 55)(5 52 7 50)(6 51 8 49)(9 60 11 58)(10 59 12 57)(13 61 15 63)(14 64 16 62)(17 68 19 66)(18 67 20 65)(21 69 23 71)(22 72 24 70)(25 76 27 74)(26 75 28 73)(29 77 31 79)(30 80 32 78)(33 84 35 82)(34 83 36 81)(37 85 39 87)(38 88 40 86)(41 92 43 90)(42 91 44 89)(45 93 47 95)(46 96 48 94)
G:=sub<Sym(96)| (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,35)(2,34)(3,33)(4,36)(5,70)(6,69)(7,72)(8,71)(9,29)(10,32)(11,31)(12,30)(13,42)(14,41)(15,44)(16,43)(17,37)(18,40)(19,39)(20,38)(21,50)(22,49)(23,52)(24,51)(25,48)(26,47)(27,46)(28,45)(53,84)(54,83)(55,82)(56,81)(57,77)(58,80)(59,79)(60,78)(61,92)(62,91)(63,90)(64,89)(65,85)(66,88)(67,87)(68,86)(73,94)(74,93)(75,96)(76,95), (1,59,11,55)(2,58,12,54)(3,57,9,53)(4,60,10,56)(5,49,96,45)(6,52,93,48)(7,51,94,47)(8,50,95,46)(13,66,20,62)(14,65,17,61)(15,68,18,64)(16,67,19,63)(21,75,27,70)(22,74,28,69)(23,73,25,72)(24,76,26,71)(29,83,33,80)(30,82,34,79)(31,81,35,78)(32,84,36,77)(37,91,41,88)(38,90,42,87)(39,89,43,86)(40,92,44,85), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,60,11,58)(10,59,12,57)(13,61,15,63)(14,64,16,62)(17,68,19,66)(18,67,20,65)(21,69,23,71)(22,72,24,70)(25,76,27,74)(26,75,28,73)(29,77,31,79)(30,80,32,78)(33,84,35,82)(34,83,36,81)(37,85,39,87)(38,88,40,86)(41,92,43,90)(42,91,44,89)(45,93,47,95)(46,96,48,94)>;
G:=Group( (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,91,83)(6,92,84)(7,89,81)(8,90,82)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,74,65)(58,75,66)(59,76,67)(60,73,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,35)(2,34)(3,33)(4,36)(5,70)(6,69)(7,72)(8,71)(9,29)(10,32)(11,31)(12,30)(13,42)(14,41)(15,44)(16,43)(17,37)(18,40)(19,39)(20,38)(21,50)(22,49)(23,52)(24,51)(25,48)(26,47)(27,46)(28,45)(53,84)(54,83)(55,82)(56,81)(57,77)(58,80)(59,79)(60,78)(61,92)(62,91)(63,90)(64,89)(65,85)(66,88)(67,87)(68,86)(73,94)(74,93)(75,96)(76,95), (1,59,11,55)(2,58,12,54)(3,57,9,53)(4,60,10,56)(5,49,96,45)(6,52,93,48)(7,51,94,47)(8,50,95,46)(13,66,20,62)(14,65,17,61)(15,68,18,64)(16,67,19,63)(21,75,27,70)(22,74,28,69)(23,73,25,72)(24,76,26,71)(29,83,33,80)(30,82,34,79)(31,81,35,78)(32,84,36,77)(37,91,41,88)(38,90,42,87)(39,89,43,86)(40,92,44,85), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,60,11,58)(10,59,12,57)(13,61,15,63)(14,64,16,62)(17,68,19,66)(18,67,20,65)(21,69,23,71)(22,72,24,70)(25,76,27,74)(26,75,28,73)(29,77,31,79)(30,80,32,78)(33,84,35,82)(34,83,36,81)(37,85,39,87)(38,88,40,86)(41,92,43,90)(42,91,44,89)(45,93,47,95)(46,96,48,94) );
G=PermutationGroup([[(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,91,83),(6,92,84),(7,89,81),(8,90,82),(9,28,17),(10,25,18),(11,26,19),(12,27,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,49,41),(34,50,42),(35,51,43),(36,52,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,74,65),(58,75,66),(59,76,67),(60,73,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(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)], [(1,35),(2,34),(3,33),(4,36),(5,70),(6,69),(7,72),(8,71),(9,29),(10,32),(11,31),(12,30),(13,42),(14,41),(15,44),(16,43),(17,37),(18,40),(19,39),(20,38),(21,50),(22,49),(23,52),(24,51),(25,48),(26,47),(27,46),(28,45),(53,84),(54,83),(55,82),(56,81),(57,77),(58,80),(59,79),(60,78),(61,92),(62,91),(63,90),(64,89),(65,85),(66,88),(67,87),(68,86),(73,94),(74,93),(75,96),(76,95)], [(1,59,11,55),(2,58,12,54),(3,57,9,53),(4,60,10,56),(5,49,96,45),(6,52,93,48),(7,51,94,47),(8,50,95,46),(13,66,20,62),(14,65,17,61),(15,68,18,64),(16,67,19,63),(21,75,27,70),(22,74,28,69),(23,73,25,72),(24,76,26,71),(29,83,33,80),(30,82,34,79),(31,81,35,78),(32,84,36,77),(37,91,41,88),(38,90,42,87),(39,89,43,86),(40,92,44,85)], [(1,54,3,56),(2,53,4,55),(5,52,7,50),(6,51,8,49),(9,60,11,58),(10,59,12,57),(13,61,15,63),(14,64,16,62),(17,68,19,66),(18,67,20,65),(21,69,23,71),(22,72,24,70),(25,76,27,74),(26,75,28,73),(29,77,31,79),(30,80,32,78),(33,84,35,82),(34,83,36,81),(37,85,39,87),(38,88,40,86),(41,92,43,90),(42,91,44,89),(45,93,47,95),(46,96,48,94)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | D4 | C3×D4 | C3×D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D8 | C8.C22 | C3×C8.C22 |
| kernel | C3×D4.7D4 | C3×C22⋊C8 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C22⋊Q8 | C6×SD16 | C6×Q16 | C6×C4○D4 | D4.7D4 | C22⋊C8 | D4⋊C4 | Q8⋊C4 | C22⋊Q8 | C2×SD16 | C2×Q16 | C2×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C22×C6 | C2×C4 | D4 | Q8 | C23 | C6 | C2 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 2 |
Matrix representation of C3×D4.7D4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 46 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 63 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 51 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,27,0,0,0,0,46],[72,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[0,1,0,0,72,0,0,0,0,0,0,63,0,0,51,0],[0,1,0,0,1,0,0,0,0,0,0,51,0,0,10,0] >;
C3×D4.7D4 in GAP, Magma, Sage, TeX
C_3\times D_4._7D_4
% in TeX
G:=Group("C3xD4.7D4"); // GroupNames label
G:=SmallGroup(192,885);
// by ID
G=gap.SmallGroup(192,885);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,520,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations