direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.46C24, C6.1202- 1+4, (C4×Q8)⋊18C6, (Q8×C12)⋊34C2, (C4×D4).11C6, C22⋊Q8⋊16C6, C42.C2⋊8C6, C42⋊2C2⋊7C6, (D4×C12).26C2, C42.46(C2×C6), C42⋊C2⋊15C6, (C2×C6).372C24, C12.323(C4○D4), (C2×C12).679C23, (C4×C12).287C22, (C6×D4).324C22, C23.48(C22×C6), C22.46(C23×C6), (C6×Q8).276C22, C22.D4.2C6, (C22×C6).267C23, C2.12(C3×2- 1+4), (C22×C12).457C22, (C2×C4⋊C4)⋊21C6, (C6×C4⋊C4)⋊48C2, C4⋊C4.33(C2×C6), C4.35(C3×C4○D4), C2.25(C6×C4○D4), (C2×D4).70(C2×C6), C6.244(C2×C4○D4), (C3×C22⋊Q8)⋊43C2, (C2×Q8).75(C2×C6), C22⋊C4.22(C2×C6), (C22×C4).74(C2×C6), (C2×C4).35(C22×C6), (C3×C42.C2)⋊25C2, C22.10(C3×C4○D4), (C3×C42⋊C2)⋊36C2, (C3×C42⋊2C2)⋊16C2, (C2×C6).119(C4○D4), (C3×C4⋊C4).250C22, (C3×C22⋊C4).89C22, (C3×C22.D4).5C2, SmallGroup(192,1441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.46C24
G = < a,b,c,d,e,f,g | a3=b2=c2=g2=1, d2=f2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 282 in 214 conjugacy classes, 150 normal (62 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, 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×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, 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×Q8, C22.46C24, C6×C4⋊C4, C3×C42⋊C2, C3×C42⋊C2, D4×C12, Q8×C12, C3×C22⋊Q8, C3×C22.D4, C3×C42.C2, C3×C42.C2, C3×C42⋊2C2, C3×C22.46C24
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, C22.46C24, C6×C4○D4, C3×2- 1+4, C3×C22.46C24
►On 96 points
Generators in S96
(1 38 51)(2 39 52)(3 40 49)(4 37 50)(5 83 91)(6 84 92)(7 81 89)(8 82 90)(9 88 95)(10 85 96)(11 86 93)(12 87 94)(13 69 73)(14 70 74)(15 71 75)(16 72 76)(17 21 77)(18 22 78)(19 23 79)(20 24 80)(25 58 62)(26 59 63)(27 60 64)(28 57 61)(29 33 66)(30 34 67)(31 35 68)(32 36 65)(41 46 53)(42 47 54)(43 48 55)(44 45 56)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)
(1 11)(2 12)(3 9)(4 10)(5 48)(6 45)(7 46)(8 47)(13 66)(14 67)(15 68)(16 65)(17 60)(18 57)(19 58)(20 59)(21 64)(22 61)(23 62)(24 63)(25 79)(26 80)(27 77)(28 78)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(49 95)(50 96)(51 93)(52 94)(53 81)(54 82)(55 83)(56 84)
(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 70 11 30)(2 69 12 29)(3 72 9 32)(4 71 10 31)(5 28 48 78)(6 27 45 77)(7 26 46 80)(8 25 47 79)(13 94 66 52)(14 93 67 51)(15 96 68 50)(16 95 65 49)(17 84 60 56)(18 83 57 55)(19 82 58 54)(20 81 59 53)(21 92 64 44)(22 91 61 43)(23 90 62 42)(24 89 63 41)(33 39 73 87)(34 38 74 86)(35 37 75 85)(36 40 76 88)
(1 17 3 19)(2 57 4 59)(5 68 7 66)(6 16 8 14)(9 58 11 60)(10 20 12 18)(13 48 15 46)(21 40 23 38)(22 85 24 87)(25 93 27 95)(26 52 28 50)(29 83 31 81)(30 56 32 54)(33 91 35 89)(34 44 36 42)(37 63 39 61)(41 73 43 75)(45 65 47 67)(49 79 51 77)(53 69 55 71)(62 86 64 88)(70 84 72 82)(74 92 76 90)(78 96 80 94)
(1 55)(2 56)(3 53)(4 54)(5 93)(6 94)(7 95)(8 96)(9 81)(10 82)(11 83)(12 84)(13 79)(14 80)(15 77)(16 78)(17 71)(18 72)(19 69)(20 70)(21 75)(22 76)(23 73)(24 74)(25 66)(26 67)(27 68)(28 65)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 61)(37 42)(38 43)(39 44)(40 41)(45 52)(46 49)(47 50)(48 51)(85 90)(86 91)(87 92)(88 89)
G:=sub<Sym(96)| (1,38,51)(2,39,52)(3,40,49)(4,37,50)(5,83,91)(6,84,92)(7,81,89)(8,82,90)(9,88,95)(10,85,96)(11,86,93)(12,87,94)(13,69,73)(14,70,74)(15,71,75)(16,72,76)(17,21,77)(18,22,78)(19,23,79)(20,24,80)(25,58,62)(26,59,63)(27,60,64)(28,57,61)(29,33,66)(30,34,67)(31,35,68)(32,36,65)(41,46,53)(42,47,54)(43,48,55)(44,45,56), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96), (1,11)(2,12)(3,9)(4,10)(5,48)(6,45)(7,46)(8,47)(13,66)(14,67)(15,68)(16,65)(17,60)(18,57)(19,58)(20,59)(21,64)(22,61)(23,62)(24,63)(25,79)(26,80)(27,77)(28,78)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(49,95)(50,96)(51,93)(52,94)(53,81)(54,82)(55,83)(56,84), (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), (1,70,11,30)(2,69,12,29)(3,72,9,32)(4,71,10,31)(5,28,48,78)(6,27,45,77)(7,26,46,80)(8,25,47,79)(13,94,66,52)(14,93,67,51)(15,96,68,50)(16,95,65,49)(17,84,60,56)(18,83,57,55)(19,82,58,54)(20,81,59,53)(21,92,64,44)(22,91,61,43)(23,90,62,42)(24,89,63,41)(33,39,73,87)(34,38,74,86)(35,37,75,85)(36,40,76,88), (1,17,3,19)(2,57,4,59)(5,68,7,66)(6,16,8,14)(9,58,11,60)(10,20,12,18)(13,48,15,46)(21,40,23,38)(22,85,24,87)(25,93,27,95)(26,52,28,50)(29,83,31,81)(30,56,32,54)(33,91,35,89)(34,44,36,42)(37,63,39,61)(41,73,43,75)(45,65,47,67)(49,79,51,77)(53,69,55,71)(62,86,64,88)(70,84,72,82)(74,92,76,90)(78,96,80,94), (1,55)(2,56)(3,53)(4,54)(5,93)(6,94)(7,95)(8,96)(9,81)(10,82)(11,83)(12,84)(13,79)(14,80)(15,77)(16,78)(17,71)(18,72)(19,69)(20,70)(21,75)(22,76)(23,73)(24,74)(25,66)(26,67)(27,68)(28,65)(29,58)(30,59)(31,60)(32,57)(33,62)(34,63)(35,64)(36,61)(37,42)(38,43)(39,44)(40,41)(45,52)(46,49)(47,50)(48,51)(85,90)(86,91)(87,92)(88,89)>;
G:=Group( (1,38,51)(2,39,52)(3,40,49)(4,37,50)(5,83,91)(6,84,92)(7,81,89)(8,82,90)(9,88,95)(10,85,96)(11,86,93)(12,87,94)(13,69,73)(14,70,74)(15,71,75)(16,72,76)(17,21,77)(18,22,78)(19,23,79)(20,24,80)(25,58,62)(26,59,63)(27,60,64)(28,57,61)(29,33,66)(30,34,67)(31,35,68)(32,36,65)(41,46,53)(42,47,54)(43,48,55)(44,45,56), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96), (1,11)(2,12)(3,9)(4,10)(5,48)(6,45)(7,46)(8,47)(13,66)(14,67)(15,68)(16,65)(17,60)(18,57)(19,58)(20,59)(21,64)(22,61)(23,62)(24,63)(25,79)(26,80)(27,77)(28,78)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(49,95)(50,96)(51,93)(52,94)(53,81)(54,82)(55,83)(56,84), (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), (1,70,11,30)(2,69,12,29)(3,72,9,32)(4,71,10,31)(5,28,48,78)(6,27,45,77)(7,26,46,80)(8,25,47,79)(13,94,66,52)(14,93,67,51)(15,96,68,50)(16,95,65,49)(17,84,60,56)(18,83,57,55)(19,82,58,54)(20,81,59,53)(21,92,64,44)(22,91,61,43)(23,90,62,42)(24,89,63,41)(33,39,73,87)(34,38,74,86)(35,37,75,85)(36,40,76,88), (1,17,3,19)(2,57,4,59)(5,68,7,66)(6,16,8,14)(9,58,11,60)(10,20,12,18)(13,48,15,46)(21,40,23,38)(22,85,24,87)(25,93,27,95)(26,52,28,50)(29,83,31,81)(30,56,32,54)(33,91,35,89)(34,44,36,42)(37,63,39,61)(41,73,43,75)(45,65,47,67)(49,79,51,77)(53,69,55,71)(62,86,64,88)(70,84,72,82)(74,92,76,90)(78,96,80,94), (1,55)(2,56)(3,53)(4,54)(5,93)(6,94)(7,95)(8,96)(9,81)(10,82)(11,83)(12,84)(13,79)(14,80)(15,77)(16,78)(17,71)(18,72)(19,69)(20,70)(21,75)(22,76)(23,73)(24,74)(25,66)(26,67)(27,68)(28,65)(29,58)(30,59)(31,60)(32,57)(33,62)(34,63)(35,64)(36,61)(37,42)(38,43)(39,44)(40,41)(45,52)(46,49)(47,50)(48,51)(85,90)(86,91)(87,92)(88,89) );
G=PermutationGroup([[(1,38,51),(2,39,52),(3,40,49),(4,37,50),(5,83,91),(6,84,92),(7,81,89),(8,82,90),(9,88,95),(10,85,96),(11,86,93),(12,87,94),(13,69,73),(14,70,74),(15,71,75),(16,72,76),(17,21,77),(18,22,78),(19,23,79),(20,24,80),(25,58,62),(26,59,63),(27,60,64),(28,57,61),(29,33,66),(30,34,67),(31,35,68),(32,36,65),(41,46,53),(42,47,54),(43,48,55),(44,45,56)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96)], [(1,11),(2,12),(3,9),(4,10),(5,48),(6,45),(7,46),(8,47),(13,66),(14,67),(15,68),(16,65),(17,60),(18,57),(19,58),(20,59),(21,64),(22,61),(23,62),(24,63),(25,79),(26,80),(27,77),(28,78),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(49,95),(50,96),(51,93),(52,94),(53,81),(54,82),(55,83),(56,84)], [(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)], [(1,70,11,30),(2,69,12,29),(3,72,9,32),(4,71,10,31),(5,28,48,78),(6,27,45,77),(7,26,46,80),(8,25,47,79),(13,94,66,52),(14,93,67,51),(15,96,68,50),(16,95,65,49),(17,84,60,56),(18,83,57,55),(19,82,58,54),(20,81,59,53),(21,92,64,44),(22,91,61,43),(23,90,62,42),(24,89,63,41),(33,39,73,87),(34,38,74,86),(35,37,75,85),(36,40,76,88)], [(1,17,3,19),(2,57,4,59),(5,68,7,66),(6,16,8,14),(9,58,11,60),(10,20,12,18),(13,48,15,46),(21,40,23,38),(22,85,24,87),(25,93,27,95),(26,52,28,50),(29,83,31,81),(30,56,32,54),(33,91,35,89),(34,44,36,42),(37,63,39,61),(41,73,43,75),(45,65,47,67),(49,79,51,77),(53,69,55,71),(62,86,64,88),(70,84,72,82),(74,92,76,90),(78,96,80,94)], [(1,55),(2,56),(3,53),(4,54),(5,93),(6,94),(7,95),(8,96),(9,81),(10,82),(11,83),(12,84),(13,79),(14,80),(15,77),(16,78),(17,71),(18,72),(19,69),(20,70),(21,75),(22,76),(23,73),(24,74),(25,66),(26,67),(27,68),(28,65),(29,58),(30,59),(31,60),(32,57),(33,62),(34,63),(35,64),(36,61),(37,42),(38,43),(39,44),(40,41),(45,52),(46,49),(47,50),(48,51),(85,90),(86,91),(87,92),(88,89)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | ··· | 4J | 4K | ··· | 4R | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 12A | ··· | 12T | 12U | ··· | 12AJ |
| order | 1 | 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 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | 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×C22.46C24 | C6×C4⋊C4 | C3×C42⋊C2 | D4×C12 | Q8×C12 | C3×C22⋊Q8 | C3×C22.D4 | C3×C42.C2 | C3×C42⋊2C2 | C22.46C24 | C2×C4⋊C4 | C42⋊C2 | C4×D4 | C4×Q8 | C22⋊Q8 | C22.D4 | C42.C2 | C42⋊2C2 | C12 | C2×C6 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 3 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 6 | 2 | 2 | 4 | 4 | 6 | 4 | 4 | 4 | 8 | 8 | 1 | 2 |
Matrix representation of C3×C22.46C24 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 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 |
| 0 | 5 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 8 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[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],[0,5,0,0,5,0,0,0,0,0,8,8,0,0,10,5],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,11,12],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12] >;
C3×C22.46C24 in GAP, Magma, Sage, TeX
C_3\times C_2^2._{46}C_2^4 % in TeX
G:=Group("C3xC2^2.46C2^4"); // GroupNames label
G:=SmallGroup(192,1441);
// by ID
G=gap.SmallGroup(192,1441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,680,2102,268,794]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=g^2=1,d^2=f^2=b,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=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