direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.47C24, C6.1642+ 1+4, (C4×D4)⋊20C6, (D4×C12)⋊49C2, C4⋊D4⋊15C6, C42.C2⋊9C6, C42⋊2C2⋊8C6, C42.47(C2×C6), C42⋊C2⋊16C6, (C2×C6).373C24, C12.324(C4○D4), (C2×C12).963C23, (C4×C12).288C22, (C6×D4).221C22, C22.D4⋊11C6, C23.19(C22×C6), C22.47(C23×C6), (C22×C6).102C23, C2.16(C3×2+ 1+4), (C22×C12).458C22, (C6×C4⋊C4)⋊49C2, (C2×C4⋊C4)⋊22C6, C4⋊C4.73(C2×C6), C4.36(C3×C4○D4), C2.26(C6×C4○D4), (C3×C4⋊D4)⋊42C2, (C2×D4).34(C2×C6), C6.245(C2×C4○D4), C22⋊C4.23(C2×C6), (C22×C4).17(C2×C6), (C2×C4).62(C22×C6), (C3×C42.C2)⋊26C2, C22.11(C3×C4○D4), (C3×C42⋊C2)⋊37C2, (C3×C42⋊2C2)⋊17C2, (C2×C6).179(C4○D4), (C3×C4⋊C4).399C22, (C3×C22.D4)⋊30C2, (C3×C22⋊C4).155C22, SmallGroup(192,1442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.47C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 362 in 238 conjugacy classes, 150 normal (62 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×C12, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C6×D4, C22.47C24, C6×C4⋊C4, C3×C42⋊C2, D4×C12, D4×C12, C3×C4⋊D4, C3×C4⋊D4, C3×C22.D4, C3×C42.C2, C3×C42⋊2C2, C3×C22.47C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, 2+ 1+4, C3×C4○D4, C23×C6, C22.47C24, C6×C4○D4, C3×2+ 1+4, C3×C22.47C24
►On 96 points
Generators in S96
(1 79 31)(2 80 32)(3 77 29)(4 78 30)(5 9 53)(6 10 54)(7 11 55)(8 12 56)(13 57 61)(14 58 62)(15 59 63)(16 60 64)(17 21 65)(18 22 66)(19 23 67)(20 24 68)(25 69 73)(26 70 74)(27 71 75)(28 72 76)(33 37 81)(34 38 82)(35 39 83)(36 40 84)(41 85 89)(42 86 90)(43 87 91)(44 88 92)(45 49 93)(46 50 94)(47 51 95)(48 52 96)
(1 5)(2 6)(3 7)(4 8)(9 79)(10 80)(11 77)(12 78)(13 83)(14 84)(15 81)(16 82)(17 87)(18 88)(19 85)(20 86)(21 91)(22 92)(23 89)(24 90)(25 95)(26 96)(27 93)(28 94)(29 55)(30 56)(31 53)(32 54)(33 59)(34 60)(35 57)(36 58)(37 63)(38 64)(39 61)(40 62)(41 67)(42 68)(43 65)(44 66)(45 71)(46 72)(47 69)(48 70)(49 75)(50 76)(51 73)(52 74)
(1 7)(2 8)(3 5)(4 6)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)
(1 86)(2 17)(3 88)(4 19)(5 20)(6 87)(7 18)(8 85)(9 24)(10 91)(11 22)(12 89)(13 94)(14 25)(15 96)(16 27)(21 80)(23 78)(26 81)(28 83)(29 44)(30 67)(31 42)(32 65)(33 70)(34 45)(35 72)(36 47)(37 74)(38 49)(39 76)(40 51)(41 56)(43 54)(46 57)(48 59)(50 61)(52 63)(53 68)(55 66)(58 69)(60 71)(62 73)(64 75)(77 92)(79 90)(82 93)(84 95)
(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 87)(2 88)(3 85)(4 86)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 57 5 35)(2 36 6 58)(3 59 7 33)(4 34 8 60)(9 39 79 61)(10 62 80 40)(11 37 77 63)(12 64 78 38)(13 53 83 31)(14 32 84 54)(15 55 81 29)(16 30 82 56)(17 47 87 69)(18 70 88 48)(19 45 85 71)(20 72 86 46)(21 51 91 73)(22 74 92 52)(23 49 89 75)(24 76 90 50)(25 65 95 43)(26 44 96 66)(27 67 93 41)(28 42 94 68)
G:=sub<Sym(96)| (1,79,31)(2,80,32)(3,77,29)(4,78,30)(5,9,53)(6,10,54)(7,11,55)(8,12,56)(13,57,61)(14,58,62)(15,59,63)(16,60,64)(17,21,65)(18,22,66)(19,23,67)(20,24,68)(25,69,73)(26,70,74)(27,71,75)(28,72,76)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,77)(12,78)(13,83)(14,84)(15,81)(16,82)(17,87)(18,88)(19,85)(20,86)(21,91)(22,92)(23,89)(24,90)(25,95)(26,96)(27,93)(28,94)(29,55)(30,56)(31,53)(32,54)(33,59)(34,60)(35,57)(36,58)(37,63)(38,64)(39,61)(40,62)(41,67)(42,68)(43,65)(44,66)(45,71)(46,72)(47,69)(48,70)(49,75)(50,76)(51,73)(52,74), (1,7)(2,8)(3,5)(4,6)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76), (1,86)(2,17)(3,88)(4,19)(5,20)(6,87)(7,18)(8,85)(9,24)(10,91)(11,22)(12,89)(13,94)(14,25)(15,96)(16,27)(21,80)(23,78)(26,81)(28,83)(29,44)(30,67)(31,42)(32,65)(33,70)(34,45)(35,72)(36,47)(37,74)(38,49)(39,76)(40,51)(41,56)(43,54)(46,57)(48,59)(50,61)(52,63)(53,68)(55,66)(58,69)(60,71)(62,73)(64,75)(77,92)(79,90)(82,93)(84,95), (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), (1,87)(2,88)(3,85)(4,86)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,57,5,35)(2,36,6,58)(3,59,7,33)(4,34,8,60)(9,39,79,61)(10,62,80,40)(11,37,77,63)(12,64,78,38)(13,53,83,31)(14,32,84,54)(15,55,81,29)(16,30,82,56)(17,47,87,69)(18,70,88,48)(19,45,85,71)(20,72,86,46)(21,51,91,73)(22,74,92,52)(23,49,89,75)(24,76,90,50)(25,65,95,43)(26,44,96,66)(27,67,93,41)(28,42,94,68)>;
G:=Group( (1,79,31)(2,80,32)(3,77,29)(4,78,30)(5,9,53)(6,10,54)(7,11,55)(8,12,56)(13,57,61)(14,58,62)(15,59,63)(16,60,64)(17,21,65)(18,22,66)(19,23,67)(20,24,68)(25,69,73)(26,70,74)(27,71,75)(28,72,76)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,77)(12,78)(13,83)(14,84)(15,81)(16,82)(17,87)(18,88)(19,85)(20,86)(21,91)(22,92)(23,89)(24,90)(25,95)(26,96)(27,93)(28,94)(29,55)(30,56)(31,53)(32,54)(33,59)(34,60)(35,57)(36,58)(37,63)(38,64)(39,61)(40,62)(41,67)(42,68)(43,65)(44,66)(45,71)(46,72)(47,69)(48,70)(49,75)(50,76)(51,73)(52,74), (1,7)(2,8)(3,5)(4,6)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76), (1,86)(2,17)(3,88)(4,19)(5,20)(6,87)(7,18)(8,85)(9,24)(10,91)(11,22)(12,89)(13,94)(14,25)(15,96)(16,27)(21,80)(23,78)(26,81)(28,83)(29,44)(30,67)(31,42)(32,65)(33,70)(34,45)(35,72)(36,47)(37,74)(38,49)(39,76)(40,51)(41,56)(43,54)(46,57)(48,59)(50,61)(52,63)(53,68)(55,66)(58,69)(60,71)(62,73)(64,75)(77,92)(79,90)(82,93)(84,95), (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), (1,87)(2,88)(3,85)(4,86)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,57,5,35)(2,36,6,58)(3,59,7,33)(4,34,8,60)(9,39,79,61)(10,62,80,40)(11,37,77,63)(12,64,78,38)(13,53,83,31)(14,32,84,54)(15,55,81,29)(16,30,82,56)(17,47,87,69)(18,70,88,48)(19,45,85,71)(20,72,86,46)(21,51,91,73)(22,74,92,52)(23,49,89,75)(24,76,90,50)(25,65,95,43)(26,44,96,66)(27,67,93,41)(28,42,94,68) );
G=PermutationGroup([[(1,79,31),(2,80,32),(3,77,29),(4,78,30),(5,9,53),(6,10,54),(7,11,55),(8,12,56),(13,57,61),(14,58,62),(15,59,63),(16,60,64),(17,21,65),(18,22,66),(19,23,67),(20,24,68),(25,69,73),(26,70,74),(27,71,75),(28,72,76),(33,37,81),(34,38,82),(35,39,83),(36,40,84),(41,85,89),(42,86,90),(43,87,91),(44,88,92),(45,49,93),(46,50,94),(47,51,95),(48,52,96)], [(1,5),(2,6),(3,7),(4,8),(9,79),(10,80),(11,77),(12,78),(13,83),(14,84),(15,81),(16,82),(17,87),(18,88),(19,85),(20,86),(21,91),(22,92),(23,89),(24,90),(25,95),(26,96),(27,93),(28,94),(29,55),(30,56),(31,53),(32,54),(33,59),(34,60),(35,57),(36,58),(37,63),(38,64),(39,61),(40,62),(41,67),(42,68),(43,65),(44,66),(45,71),(46,72),(47,69),(48,70),(49,75),(50,76),(51,73),(52,74)], [(1,7),(2,8),(3,5),(4,6),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76)], [(1,86),(2,17),(3,88),(4,19),(5,20),(6,87),(7,18),(8,85),(9,24),(10,91),(11,22),(12,89),(13,94),(14,25),(15,96),(16,27),(21,80),(23,78),(26,81),(28,83),(29,44),(30,67),(31,42),(32,65),(33,70),(34,45),(35,72),(36,47),(37,74),(38,49),(39,76),(40,51),(41,56),(43,54),(46,57),(48,59),(50,61),(52,63),(53,68),(55,66),(58,69),(60,71),(62,73),(64,75),(77,92),(79,90),(82,93),(84,95)], [(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)], [(1,87),(2,88),(3,85),(4,86),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,57,5,35),(2,36,6,58),(3,59,7,33),(4,34,8,60),(9,39,79,61),(10,62,80,40),(11,37,77,63),(12,64,78,38),(13,53,83,31),(14,32,84,54),(15,55,81,29),(16,30,82,56),(17,47,87,69),(18,70,88,48),(19,45,85,71),(20,72,86,46),(21,51,91,73),(22,74,92,52),(23,49,89,75),(24,76,90,50),(25,65,95,43),(26,44,96,66),(27,67,93,41),(28,42,94,68)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3A | 3B | 4A | ··· | 4J | 4K | ··· | 4P | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6P | 12A | ··· | 12T | 12U | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C4○D4 | C4○D4 | C3×C4○D4 | C3×C4○D4 | 2+ 1+4 | C3×2+ 1+4 |
| kernel | C3×C22.47C24 | C6×C4⋊C4 | C3×C42⋊C2 | D4×C12 | C3×C4⋊D4 | C3×C22.D4 | C3×C42.C2 | C3×C42⋊2C2 | C22.47C24 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4⋊D4 | C22.D4 | C42.C2 | C42⋊2C2 | C12 | C2×C6 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 2 |
Matrix representation of C3×C22.47C24 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 10 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 8 | 5 |
| 8 | 3 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 12 |
| 1 | 2 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,10,8,0,0,0,0,8,8,0,0,10,5],[8,0,0,0,3,5,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,11,12],[1,12,0,0,2,12,0,0,0,0,1,0,0,0,0,1] >;
C3×C22.47C24 in GAP, Magma, Sage, TeX
C_3\times C_2^2._{47}C_2^4 % in TeX
G:=Group("C3xC2^2.47C2^4"); // GroupNames label
G:=SmallGroup(192,1442);
// by ID
G=gap.SmallGroup(192,1442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,1016,2102,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations