direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.50C24, C6.1212- 1+4, C4⋊Q8⋊17C6, (C4×Q8)⋊20C6, (Q8×C12)⋊36C2, (C4×D4).13C6, C22⋊Q8⋊18C6, C42⋊2C2⋊9C6, (D4×C12).28C2, C42.50(C2×C6), C4.4D4.8C6, C42⋊C2⋊18C6, (C2×C6).376C24, C12.347(C4○D4), (C2×C12).965C23, (C4×C12).291C22, (C6×D4).325C22, C23.21(C22×C6), C22.50(C23×C6), (C6×Q8).277C22, (C22×C6).104C23, C2.13(C3×2- 1+4), (C22×C12).461C22, (C3×C4⋊Q8)⋊38C2, C4⋊C4.76(C2×C6), C4.39(C3×C4○D4), C2.29(C6×C4○D4), (C2×D4).71(C2×C6), C6.248(C2×C4○D4), C22⋊C4.6(C2×C6), (C3×C22⋊Q8)⋊45C2, (C2×Q8).77(C2×C6), (C2×C4).38(C22×C6), (C22×C4).77(C2×C6), (C3×C42⋊2C2)⋊18C2, (C3×C42⋊C2)⋊39C2, (C3×C4⋊C4).401C22, (C3×C4.4D4).17C2, (C3×C22⋊C4).90C22, SmallGroup(192,1445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.50C24
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=cb=bc, f2=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-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 282 in 212 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×Q8, C22.50C24, C3×C42⋊C2, D4×C12, Q8×C12, Q8×C12, C3×C22⋊Q8, C3×C4.4D4, C3×C42⋊2C2, C3×C4⋊Q8, C3×C22.50C24
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, C22.50C24, C6×C4○D4, C3×2- 1+4, C3×C22.50C24
►On 96 points
Generators in S96
(1 43 19)(2 44 20)(3 41 17)(4 42 18)(5 9 29)(6 10 30)(7 11 31)(8 12 32)(13 33 37)(14 34 38)(15 35 39)(16 36 40)(21 51 45)(22 52 46)(23 49 47)(24 50 48)(25 93 71)(26 94 72)(27 95 69)(28 96 70)(53 75 79)(54 76 80)(55 73 77)(56 74 78)(57 61 83)(58 62 84)(59 63 81)(60 64 82)(65 87 91)(66 88 92)(67 85 89)(68 86 90)
(1 5)(2 6)(3 7)(4 8)(9 43)(10 44)(11 41)(12 42)(13 47)(14 48)(15 45)(16 46)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)(25 62)(26 63)(27 64)(28 61)(37 49)(38 50)(39 51)(40 52)(53 66)(54 67)(55 68)(56 65)(57 70)(58 71)(59 72)(60 69)(73 86)(74 87)(75 88)(76 85)(77 90)(78 91)(79 92)(80 89)(81 94)(82 95)(83 96)(84 93)
(1 7)(2 8)(3 5)(4 6)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 64)(26 61)(27 62)(28 63)(37 51)(38 52)(39 49)(40 50)(53 68)(54 65)(55 66)(56 67)(57 72)(58 69)(59 70)(60 71)(73 88)(74 85)(75 86)(76 87)(77 92)(78 89)(79 90)(80 91)(81 96)(82 93)(83 94)(84 95)
(1 74)(2 88)(3 76)(4 86)(5 87)(6 75)(7 85)(8 73)(9 91)(10 79)(11 89)(12 77)(13 95)(14 83)(15 93)(16 81)(17 54)(18 68)(19 56)(20 66)(21 58)(22 72)(23 60)(24 70)(25 39)(26 52)(27 37)(28 50)(29 65)(30 53)(31 67)(32 55)(33 69)(34 57)(35 71)(36 59)(38 61)(40 63)(41 80)(42 90)(43 78)(44 92)(45 84)(46 94)(47 82)(48 96)(49 64)(51 62)
(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 23 5 33)(2 24 6 34)(3 21 7 35)(4 22 8 36)(9 37 43 49)(10 38 44 50)(11 39 41 51)(12 40 42 52)(13 19 47 29)(14 20 48 30)(15 17 45 31)(16 18 46 32)(25 91 62 78)(26 92 63 79)(27 89 64 80)(28 90 61 77)(53 94 66 81)(54 95 67 82)(55 96 68 83)(56 93 65 84)(57 73 70 86)(58 74 71 87)(59 75 72 88)(60 76 69 85)
(1 33 5 23)(2 24 6 34)(3 35 7 21)(4 22 8 36)(9 49 43 37)(10 38 44 50)(11 51 41 39)(12 40 42 52)(13 29 47 19)(14 20 48 30)(15 31 45 17)(16 18 46 32)(25 89 62 80)(26 77 63 90)(27 91 64 78)(28 79 61 92)(53 83 66 96)(54 93 67 84)(55 81 68 94)(56 95 65 82)(57 88 70 75)(58 76 71 85)(59 86 72 73)(60 74 69 87)
G:=sub<Sym(96)| (1,43,19)(2,44,20)(3,41,17)(4,42,18)(5,9,29)(6,10,30)(7,11,31)(8,12,32)(13,33,37)(14,34,38)(15,35,39)(16,36,40)(21,51,45)(22,52,46)(23,49,47)(24,50,48)(25,93,71)(26,94,72)(27,95,69)(28,96,70)(53,75,79)(54,76,80)(55,73,77)(56,74,78)(57,61,83)(58,62,84)(59,63,81)(60,64,82)(65,87,91)(66,88,92)(67,85,89)(68,86,90), (1,5)(2,6)(3,7)(4,8)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(25,62)(26,63)(27,64)(28,61)(37,49)(38,50)(39,51)(40,52)(53,66)(54,67)(55,68)(56,65)(57,70)(58,71)(59,72)(60,69)(73,86)(74,87)(75,88)(76,85)(77,90)(78,91)(79,92)(80,89)(81,94)(82,95)(83,96)(84,93), (1,7)(2,8)(3,5)(4,6)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,64)(26,61)(27,62)(28,63)(37,51)(38,52)(39,49)(40,50)(53,68)(54,65)(55,66)(56,67)(57,72)(58,69)(59,70)(60,71)(73,88)(74,85)(75,86)(76,87)(77,92)(78,89)(79,90)(80,91)(81,96)(82,93)(83,94)(84,95), (1,74)(2,88)(3,76)(4,86)(5,87)(6,75)(7,85)(8,73)(9,91)(10,79)(11,89)(12,77)(13,95)(14,83)(15,93)(16,81)(17,54)(18,68)(19,56)(20,66)(21,58)(22,72)(23,60)(24,70)(25,39)(26,52)(27,37)(28,50)(29,65)(30,53)(31,67)(32,55)(33,69)(34,57)(35,71)(36,59)(38,61)(40,63)(41,80)(42,90)(43,78)(44,92)(45,84)(46,94)(47,82)(48,96)(49,64)(51,62), (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), (1,23,5,33)(2,24,6,34)(3,21,7,35)(4,22,8,36)(9,37,43,49)(10,38,44,50)(11,39,41,51)(12,40,42,52)(13,19,47,29)(14,20,48,30)(15,17,45,31)(16,18,46,32)(25,91,62,78)(26,92,63,79)(27,89,64,80)(28,90,61,77)(53,94,66,81)(54,95,67,82)(55,96,68,83)(56,93,65,84)(57,73,70,86)(58,74,71,87)(59,75,72,88)(60,76,69,85), (1,33,5,23)(2,24,6,34)(3,35,7,21)(4,22,8,36)(9,49,43,37)(10,38,44,50)(11,51,41,39)(12,40,42,52)(13,29,47,19)(14,20,48,30)(15,31,45,17)(16,18,46,32)(25,89,62,80)(26,77,63,90)(27,91,64,78)(28,79,61,92)(53,83,66,96)(54,93,67,84)(55,81,68,94)(56,95,65,82)(57,88,70,75)(58,76,71,85)(59,86,72,73)(60,74,69,87)>;
G:=Group( (1,43,19)(2,44,20)(3,41,17)(4,42,18)(5,9,29)(6,10,30)(7,11,31)(8,12,32)(13,33,37)(14,34,38)(15,35,39)(16,36,40)(21,51,45)(22,52,46)(23,49,47)(24,50,48)(25,93,71)(26,94,72)(27,95,69)(28,96,70)(53,75,79)(54,76,80)(55,73,77)(56,74,78)(57,61,83)(58,62,84)(59,63,81)(60,64,82)(65,87,91)(66,88,92)(67,85,89)(68,86,90), (1,5)(2,6)(3,7)(4,8)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(25,62)(26,63)(27,64)(28,61)(37,49)(38,50)(39,51)(40,52)(53,66)(54,67)(55,68)(56,65)(57,70)(58,71)(59,72)(60,69)(73,86)(74,87)(75,88)(76,85)(77,90)(78,91)(79,92)(80,89)(81,94)(82,95)(83,96)(84,93), (1,7)(2,8)(3,5)(4,6)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,64)(26,61)(27,62)(28,63)(37,51)(38,52)(39,49)(40,50)(53,68)(54,65)(55,66)(56,67)(57,72)(58,69)(59,70)(60,71)(73,88)(74,85)(75,86)(76,87)(77,92)(78,89)(79,90)(80,91)(81,96)(82,93)(83,94)(84,95), (1,74)(2,88)(3,76)(4,86)(5,87)(6,75)(7,85)(8,73)(9,91)(10,79)(11,89)(12,77)(13,95)(14,83)(15,93)(16,81)(17,54)(18,68)(19,56)(20,66)(21,58)(22,72)(23,60)(24,70)(25,39)(26,52)(27,37)(28,50)(29,65)(30,53)(31,67)(32,55)(33,69)(34,57)(35,71)(36,59)(38,61)(40,63)(41,80)(42,90)(43,78)(44,92)(45,84)(46,94)(47,82)(48,96)(49,64)(51,62), (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), (1,23,5,33)(2,24,6,34)(3,21,7,35)(4,22,8,36)(9,37,43,49)(10,38,44,50)(11,39,41,51)(12,40,42,52)(13,19,47,29)(14,20,48,30)(15,17,45,31)(16,18,46,32)(25,91,62,78)(26,92,63,79)(27,89,64,80)(28,90,61,77)(53,94,66,81)(54,95,67,82)(55,96,68,83)(56,93,65,84)(57,73,70,86)(58,74,71,87)(59,75,72,88)(60,76,69,85), (1,33,5,23)(2,24,6,34)(3,35,7,21)(4,22,8,36)(9,49,43,37)(10,38,44,50)(11,51,41,39)(12,40,42,52)(13,29,47,19)(14,20,48,30)(15,31,45,17)(16,18,46,32)(25,89,62,80)(26,77,63,90)(27,91,64,78)(28,79,61,92)(53,83,66,96)(54,93,67,84)(55,81,68,94)(56,95,65,82)(57,88,70,75)(58,76,71,85)(59,86,72,73)(60,74,69,87) );
G=PermutationGroup([[(1,43,19),(2,44,20),(3,41,17),(4,42,18),(5,9,29),(6,10,30),(7,11,31),(8,12,32),(13,33,37),(14,34,38),(15,35,39),(16,36,40),(21,51,45),(22,52,46),(23,49,47),(24,50,48),(25,93,71),(26,94,72),(27,95,69),(28,96,70),(53,75,79),(54,76,80),(55,73,77),(56,74,78),(57,61,83),(58,62,84),(59,63,81),(60,64,82),(65,87,91),(66,88,92),(67,85,89),(68,86,90)], [(1,5),(2,6),(3,7),(4,8),(9,43),(10,44),(11,41),(12,42),(13,47),(14,48),(15,45),(16,46),(17,31),(18,32),(19,29),(20,30),(21,35),(22,36),(23,33),(24,34),(25,62),(26,63),(27,64),(28,61),(37,49),(38,50),(39,51),(40,52),(53,66),(54,67),(55,68),(56,65),(57,70),(58,71),(59,72),(60,69),(73,86),(74,87),(75,88),(76,85),(77,90),(78,91),(79,92),(80,89),(81,94),(82,95),(83,96),(84,93)], [(1,7),(2,8),(3,5),(4,6),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,64),(26,61),(27,62),(28,63),(37,51),(38,52),(39,49),(40,50),(53,68),(54,65),(55,66),(56,67),(57,72),(58,69),(59,70),(60,71),(73,88),(74,85),(75,86),(76,87),(77,92),(78,89),(79,90),(80,91),(81,96),(82,93),(83,94),(84,95)], [(1,74),(2,88),(3,76),(4,86),(5,87),(6,75),(7,85),(8,73),(9,91),(10,79),(11,89),(12,77),(13,95),(14,83),(15,93),(16,81),(17,54),(18,68),(19,56),(20,66),(21,58),(22,72),(23,60),(24,70),(25,39),(26,52),(27,37),(28,50),(29,65),(30,53),(31,67),(32,55),(33,69),(34,57),(35,71),(36,59),(38,61),(40,63),(41,80),(42,90),(43,78),(44,92),(45,84),(46,94),(47,82),(48,96),(49,64),(51,62)], [(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)], [(1,23,5,33),(2,24,6,34),(3,21,7,35),(4,22,8,36),(9,37,43,49),(10,38,44,50),(11,39,41,51),(12,40,42,52),(13,19,47,29),(14,20,48,30),(15,17,45,31),(16,18,46,32),(25,91,62,78),(26,92,63,79),(27,89,64,80),(28,90,61,77),(53,94,66,81),(54,95,67,82),(55,96,68,83),(56,93,65,84),(57,73,70,86),(58,74,71,87),(59,75,72,88),(60,76,69,85)], [(1,33,5,23),(2,24,6,34),(3,35,7,21),(4,22,8,36),(9,49,43,37),(10,38,44,50),(11,51,41,39),(12,40,42,52),(13,29,47,19),(14,20,48,30),(15,31,45,17),(16,18,46,32),(25,89,62,80),(26,77,63,90),(27,91,64,78),(28,79,61,92),(53,83,66,96),(54,93,67,84),(55,81,68,94),(56,95,65,82),(57,88,70,75),(58,76,71,85),(59,86,72,73),(60,74,69,87)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4S | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12X | 12Y | ··· | 12AL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 | 2- 1+4 | C3×2- 1+4 |
| kernel | C3×C22.50C24 | C3×C42⋊C2 | D4×C12 | Q8×C12 | C3×C22⋊Q8 | C3×C4.4D4 | C3×C42⋊2C2 | C3×C4⋊Q8 | C22.50C24 | C42⋊C2 | C4×D4 | C4×Q8 | C22⋊Q8 | C4.4D4 | C42⋊2C2 | C4⋊Q8 | C12 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 1 | 2 | 4 | 2 | 6 | 4 | 4 | 8 | 2 | 8 | 16 | 1 | 2 |
Matrix representation of C3×C22.50C24 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 2 | 0 | 0 |
| 5 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 1 | 5 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[2,5,0,0,2,11,0,0,0,0,12,8,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,8,0,0,0,2,5],[1,11,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,8,1,0,0,0,5] >;
C3×C22.50C24 in GAP, Magma, Sage, TeX
C_3\times C_2^2._{50}C_2^4 % in TeX
G:=Group("C3xC2^2.50C2^4"); // GroupNames label
G:=SmallGroup(192,1445);
// by ID
G=gap.SmallGroup(192,1445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,1016,2102,268,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=c*b=b*c,f^2=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^-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