direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.46D4, C4.Q8⋊9C6, C22⋊C8⋊10C6, C4⋊D4.6C6, D4⋊C4⋊12C6, (C2×C12).337D4, (C2×C6).36SD16, C2.12(C6×SD16), C6.92(C2×SD16), C23.51(C3×D4), (C22×C6).168D4, C22.102(C6×D4), C12.318(C4○D4), C22.8(C3×SD16), C6.142(C8⋊C22), (C2×C24).305C22, (C2×C12).937C23, (C6×D4).196C22, (C22×C12).429C22, C6.96(C22.D4), (C6×C4⋊C4)⋊39C2, (C2×C4⋊C4)⋊12C6, C4⋊C4.58(C2×C6), (C2×C8).42(C2×C6), (C3×C4.Q8)⋊24C2, C4.30(C3×C4○D4), (C2×C4).38(C3×D4), (C3×C22⋊C8)⋊27C2, (C2×D4).19(C2×C6), (C2×C6).658(C2×D4), C2.17(C3×C8⋊C22), (C3×D4⋊C4)⋊36C2, (C3×C4⋊D4).16C2, (C22×C4).52(C2×C6), (C3×C4⋊C4).381C22, (C2×C4).112(C22×C6), C2.12(C3×C22.D4), SmallGroup(192,914)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.46D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >
Subgroups: 226 in 114 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22×C12, C6×D4, C6×D4, C23.46D4, C3×C22⋊C8, C3×D4⋊C4, C3×C4.Q8, C6×C4⋊C4, C3×C4⋊D4, C3×C23.46D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C4○D4, C3×D4, C22×C6, C22.D4, C2×SD16, C8⋊C22, C3×SD16, C6×D4, C3×C4○D4, C23.46D4, C3×C22.D4, C6×SD16, C3×C8⋊C22, C3×C23.46D4
►On 96 points
Generators in S96
(1 38 12)(2 39 13)(3 40 14)(4 33 15)(5 34 16)(6 35 9)(7 36 10)(8 37 11)(17 53 43)(18 54 44)(19 55 45)(20 56 46)(21 49 47)(22 50 48)(23 51 41)(24 52 42)(25 91 69)(26 92 70)(27 93 71)(28 94 72)(29 95 65)(30 96 66)(31 89 67)(32 90 68)(57 76 82)(58 77 83)(59 78 84)(60 79 85)(61 80 86)(62 73 87)(63 74 88)(64 75 81)
(1 71)(2 88)(3 65)(4 82)(5 67)(6 84)(7 69)(8 86)(9 78)(10 91)(11 80)(12 93)(13 74)(14 95)(15 76)(16 89)(17 85)(18 70)(19 87)(20 72)(21 81)(22 66)(23 83)(24 68)(25 36)(26 54)(27 38)(28 56)(29 40)(30 50)(31 34)(32 52)(33 57)(35 59)(37 61)(39 63)(41 77)(42 90)(43 79)(44 92)(45 73)(46 94)(47 75)(48 96)(49 64)(51 58)(53 60)(55 62)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 93)(74 94)(75 95)(76 96)(77 89)(78 90)(79 91)(80 92)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 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)
(2 22)(3 7)(4 20)(6 18)(8 24)(9 44)(10 14)(11 42)(13 48)(15 46)(17 21)(25 64)(26 32)(27 62)(28 30)(29 60)(31 58)(33 56)(35 54)(36 40)(37 52)(39 50)(43 47)(49 53)(57 63)(59 61)(65 85)(66 72)(67 83)(68 70)(69 81)(71 87)(73 93)(74 76)(75 91)(77 89)(78 80)(79 95)(82 88)(84 86)(90 92)(94 96)
G:=sub<Sym(96)| (1,38,12)(2,39,13)(3,40,14)(4,33,15)(5,34,16)(6,35,9)(7,36,10)(8,37,11)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,91,69)(26,92,70)(27,93,71)(28,94,72)(29,95,65)(30,96,66)(31,89,67)(32,90,68)(57,76,82)(58,77,83)(59,78,84)(60,79,85)(61,80,86)(62,73,87)(63,74,88)(64,75,81), (1,71)(2,88)(3,65)(4,82)(5,67)(6,84)(7,69)(8,86)(9,78)(10,91)(11,80)(12,93)(13,74)(14,95)(15,76)(16,89)(17,85)(18,70)(19,87)(20,72)(21,81)(22,66)(23,83)(24,68)(25,36)(26,54)(27,38)(28,56)(29,40)(30,50)(31,34)(32,52)(33,57)(35,59)(37,61)(39,63)(41,77)(42,90)(43,79)(44,92)(45,73)(46,94)(47,75)(48,96)(49,64)(51,58)(53,60)(55,62), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,64)(26,32)(27,62)(28,30)(29,60)(31,58)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,63)(59,61)(65,85)(66,72)(67,83)(68,70)(69,81)(71,87)(73,93)(74,76)(75,91)(77,89)(78,80)(79,95)(82,88)(84,86)(90,92)(94,96)>;
G:=Group( (1,38,12)(2,39,13)(3,40,14)(4,33,15)(5,34,16)(6,35,9)(7,36,10)(8,37,11)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,91,69)(26,92,70)(27,93,71)(28,94,72)(29,95,65)(30,96,66)(31,89,67)(32,90,68)(57,76,82)(58,77,83)(59,78,84)(60,79,85)(61,80,86)(62,73,87)(63,74,88)(64,75,81), (1,71)(2,88)(3,65)(4,82)(5,67)(6,84)(7,69)(8,86)(9,78)(10,91)(11,80)(12,93)(13,74)(14,95)(15,76)(16,89)(17,85)(18,70)(19,87)(20,72)(21,81)(22,66)(23,83)(24,68)(25,36)(26,54)(27,38)(28,56)(29,40)(30,50)(31,34)(32,52)(33,57)(35,59)(37,61)(39,63)(41,77)(42,90)(43,79)(44,92)(45,73)(46,94)(47,75)(48,96)(49,64)(51,58)(53,60)(55,62), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,93)(74,94)(75,95)(76,96)(77,89)(78,90)(79,91)(80,92), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,64)(26,32)(27,62)(28,30)(29,60)(31,58)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,63)(59,61)(65,85)(66,72)(67,83)(68,70)(69,81)(71,87)(73,93)(74,76)(75,91)(77,89)(78,80)(79,95)(82,88)(84,86)(90,92)(94,96) );
G=PermutationGroup([[(1,38,12),(2,39,13),(3,40,14),(4,33,15),(5,34,16),(6,35,9),(7,36,10),(8,37,11),(17,53,43),(18,54,44),(19,55,45),(20,56,46),(21,49,47),(22,50,48),(23,51,41),(24,52,42),(25,91,69),(26,92,70),(27,93,71),(28,94,72),(29,95,65),(30,96,66),(31,89,67),(32,90,68),(57,76,82),(58,77,83),(59,78,84),(60,79,85),(61,80,86),(62,73,87),(63,74,88),(64,75,81)], [(1,71),(2,88),(3,65),(4,82),(5,67),(6,84),(7,69),(8,86),(9,78),(10,91),(11,80),(12,93),(13,74),(14,95),(15,76),(16,89),(17,85),(18,70),(19,87),(20,72),(21,81),(22,66),(23,83),(24,68),(25,36),(26,54),(27,38),(28,56),(29,40),(30,50),(31,34),(32,52),(33,57),(35,59),(37,61),(39,63),(41,77),(42,90),(43,79),(44,92),(45,73),(46,94),(47,75),(48,96),(49,64),(51,58),(53,60),(55,62)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,93),(74,94),(75,95),(76,96),(77,89),(78,90),(79,91),(80,92)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96)], [(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)], [(2,22),(3,7),(4,20),(6,18),(8,24),(9,44),(10,14),(11,42),(13,48),(15,46),(17,21),(25,64),(26,32),(27,62),(28,30),(29,60),(31,58),(33,56),(35,54),(36,40),(37,52),(39,50),(43,47),(49,53),(57,63),(59,61),(65,85),(66,72),(67,83),(68,70),(69,81),(71,87),(73,93),(74,76),(75,91),(77,89),(78,80),(79,95),(82,88),(84,86),(90,92),(94,96)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | SD16 | C3×D4 | C3×D4 | C3×C4○D4 | C3×SD16 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C23.46D4 | C3×C22⋊C8 | C3×D4⋊C4 | C3×C4.Q8 | C6×C4⋊C4 | C3×C4⋊D4 | C23.46D4 | C22⋊C8 | D4⋊C4 | C4.Q8 | C2×C4⋊C4 | C4⋊D4 | C2×C12 | C22×C6 | C12 | C2×C6 | C2×C4 | C23 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 1 | 2 |
Matrix representation of C3×C23.46D4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 67 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 27 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[6,6,0,0,67,6,0,0,0,0,27,27,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,1,1,0,0,0,72] >;
C3×C23.46D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{46}D_4 % in TeX
G:=Group("C3xC2^3.46D4"); // GroupNames label
G:=SmallGroup(192,914);
// by ID
G=gap.SmallGroup(192,914);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,646,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*e^3>;
// generators/relations