direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.36D4, C4○D4⋊4C12, D4⋊4(C2×C12), Q8⋊5(C2×C12), C4.54(C6×D4), D4⋊C4⋊14C6, Q8⋊C4⋊14C6, C12.461(C2×D4), (C2×C12).314D4, C4.4(C22×C12), C22.44(C6×D4), C23.41(C3×D4), (C2×M4(2))⋊10C6, (C6×M4(2))⋊28C2, (C22×C6).158D4, C6.126(C8⋊C22), (C2×C12).893C23, (C2×C24).322C22, C12.149(C22×C4), (C6×D4).288C22, (C6×Q8).252C22, C12.108(C22⋊C4), C6.126(C8.C22), (C22×C12).410C22, (C2×C4⋊C4)⋊10C6, (C6×C4⋊C4)⋊37C2, (C3×C4○D4)⋊8C4, (C3×D4)⋊24(C2×C4), C4⋊C4.39(C2×C6), (C2×C8).47(C2×C6), (C3×Q8)⋊22(C2×C4), C2.1(C3×C8⋊C22), (C2×C4).20(C2×C12), (C6×C4○D4).19C2, (C2×C4○D4).11C6, (C2×D4).46(C2×C6), (C2×C4).123(C3×D4), (C2×C6).620(C2×D4), C4.23(C3×C22⋊C4), C2.20(C6×C22⋊C4), (C2×Q8).49(C2×C6), (C3×D4⋊C4)⋊37C2, C2.1(C3×C8.C22), (C2×C12).193(C2×C4), (C3×Q8⋊C4)⋊37C2, C6.108(C2×C22⋊C4), (C2×C4).68(C22×C6), (C22×C4).34(C2×C6), C22.5(C3×C22⋊C4), (C3×C4⋊C4).360C22, (C2×C6).32(C22⋊C4), SmallGroup(192,850)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.36D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 274 in 162 conjugacy classes, 82 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C23.36D4, C3×D4⋊C4, C3×Q8⋊C4, C6×C4⋊C4, C6×M4(2), C6×C4○D4, C3×C23.36D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C8⋊C22, C8.C22, C3×C22⋊C4, C22×C12, C6×D4, C23.36D4, C6×C22⋊C4, C3×C8⋊C22, C3×C8.C22, C3×C23.36D4
►On 96 points
Generators in S96
(1 58 15)(2 59 16)(3 60 9)(4 61 10)(5 62 11)(6 63 12)(7 64 13)(8 57 14)(17 26 68)(18 27 69)(19 28 70)(20 29 71)(21 30 72)(22 31 65)(23 32 66)(24 25 67)(33 75 83)(34 76 84)(35 77 85)(36 78 86)(37 79 87)(38 80 88)(39 73 81)(40 74 82)(41 49 91)(42 50 92)(43 51 93)(44 52 94)(45 53 95)(46 54 96)(47 55 89)(48 56 90)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 68)(10 65)(11 70)(12 67)(13 72)(14 69)(15 66)(16 71)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(33 95)(34 92)(35 89)(36 94)(37 91)(38 96)(39 93)(40 90)(41 79)(42 76)(43 73)(44 78)(45 75)(46 80)(47 77)(48 74)(49 87)(50 84)(51 81)(52 86)(53 83)(54 88)(55 85)(56 82)
(1 79)(2 80)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 89)
(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)
(1 8 79 78)(2 77 80 7)(3 6 73 76)(4 75 74 5)(9 12 39 34)(10 33 40 11)(13 16 35 38)(14 37 36 15)(17 24 43 42)(18 41 44 23)(19 22 45 48)(20 47 46 21)(25 51 50 26)(27 49 52 32)(28 31 53 56)(29 55 54 30)(57 87 86 58)(59 85 88 64)(60 63 81 84)(61 83 82 62)(65 95 90 70)(66 69 91 94)(67 93 92 68)(71 89 96 72)
G:=sub<Sym(96)| (1,58,15)(2,59,16)(3,60,9)(4,61,10)(5,62,11)(6,63,12)(7,64,13)(8,57,14)(17,26,68)(18,27,69)(19,28,70)(20,29,71)(21,30,72)(22,31,65)(23,32,66)(24,25,67)(33,75,83)(34,76,84)(35,77,85)(36,78,86)(37,79,87)(38,80,88)(39,73,81)(40,74,82)(41,49,91)(42,50,92)(43,51,93)(44,52,94)(45,53,95)(46,54,96)(47,55,89)(48,56,90), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,68)(10,65)(11,70)(12,67)(13,72)(14,69)(15,66)(16,71)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,95)(34,92)(35,89)(36,94)(37,91)(38,96)(39,93)(40,90)(41,79)(42,76)(43,73)(44,78)(45,75)(46,80)(47,77)(48,74)(49,87)(50,84)(51,81)(52,86)(53,83)(54,88)(55,85)(56,82), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89), (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), (1,8,79,78)(2,77,80,7)(3,6,73,76)(4,75,74,5)(9,12,39,34)(10,33,40,11)(13,16,35,38)(14,37,36,15)(17,24,43,42)(18,41,44,23)(19,22,45,48)(20,47,46,21)(25,51,50,26)(27,49,52,32)(28,31,53,56)(29,55,54,30)(57,87,86,58)(59,85,88,64)(60,63,81,84)(61,83,82,62)(65,95,90,70)(66,69,91,94)(67,93,92,68)(71,89,96,72)>;
G:=Group( (1,58,15)(2,59,16)(3,60,9)(4,61,10)(5,62,11)(6,63,12)(7,64,13)(8,57,14)(17,26,68)(18,27,69)(19,28,70)(20,29,71)(21,30,72)(22,31,65)(23,32,66)(24,25,67)(33,75,83)(34,76,84)(35,77,85)(36,78,86)(37,79,87)(38,80,88)(39,73,81)(40,74,82)(41,49,91)(42,50,92)(43,51,93)(44,52,94)(45,53,95)(46,54,96)(47,55,89)(48,56,90), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,68)(10,65)(11,70)(12,67)(13,72)(14,69)(15,66)(16,71)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,95)(34,92)(35,89)(36,94)(37,91)(38,96)(39,93)(40,90)(41,79)(42,76)(43,73)(44,78)(45,75)(46,80)(47,77)(48,74)(49,87)(50,84)(51,81)(52,86)(53,83)(54,88)(55,85)(56,82), (1,79)(2,80)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89), (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), (1,8,79,78)(2,77,80,7)(3,6,73,76)(4,75,74,5)(9,12,39,34)(10,33,40,11)(13,16,35,38)(14,37,36,15)(17,24,43,42)(18,41,44,23)(19,22,45,48)(20,47,46,21)(25,51,50,26)(27,49,52,32)(28,31,53,56)(29,55,54,30)(57,87,86,58)(59,85,88,64)(60,63,81,84)(61,83,82,62)(65,95,90,70)(66,69,91,94)(67,93,92,68)(71,89,96,72) );
G=PermutationGroup([[(1,58,15),(2,59,16),(3,60,9),(4,61,10),(5,62,11),(6,63,12),(7,64,13),(8,57,14),(17,26,68),(18,27,69),(19,28,70),(20,29,71),(21,30,72),(22,31,65),(23,32,66),(24,25,67),(33,75,83),(34,76,84),(35,77,85),(36,78,86),(37,79,87),(38,80,88),(39,73,81),(40,74,82),(41,49,91),(42,50,92),(43,51,93),(44,52,94),(45,53,95),(46,54,96),(47,55,89),(48,56,90)], [(1,23),(2,20),(3,17),(4,22),(5,19),(6,24),(7,21),(8,18),(9,68),(10,65),(11,70),(12,67),(13,72),(14,69),(15,66),(16,71),(25,63),(26,60),(27,57),(28,62),(29,59),(30,64),(31,61),(32,58),(33,95),(34,92),(35,89),(36,94),(37,91),(38,96),(39,93),(40,90),(41,79),(42,76),(43,73),(44,78),(45,75),(46,80),(47,77),(48,74),(49,87),(50,84),(51,81),(52,86),(53,83),(54,88),(55,85),(56,82)], [(1,79),(2,80),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,89)], [(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)], [(1,8,79,78),(2,77,80,7),(3,6,73,76),(4,75,74,5),(9,12,39,34),(10,33,40,11),(13,16,35,38),(14,37,36,15),(17,24,43,42),(18,41,44,23),(19,22,45,48),(20,47,46,21),(25,51,50,26),(27,49,52,32),(28,31,53,56),(29,55,54,30),(57,87,86,58),(59,85,88,64),(60,63,81,84),(61,83,82,62),(65,95,90,70),(66,69,91,94),(67,93,92,68),(71,89,96,72)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D4 | C3×D4 | C3×D4 | C8⋊C22 | C8.C22 | C3×C8⋊C22 | C3×C8.C22 |
| kernel | C3×C23.36D4 | C3×D4⋊C4 | C3×Q8⋊C4 | C6×C4⋊C4 | C6×M4(2) | C6×C4○D4 | C23.36D4 | C3×C4○D4 | D4⋊C4 | Q8⋊C4 | C2×C4⋊C4 | C2×M4(2) | C2×C4○D4 | C4○D4 | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 3 | 1 | 6 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C23.36D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 71 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 22 | 4 | 0 | 0 | 0 | 0 |
| 7 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 68 | 7 | 5 |
| 0 | 0 | 66 | 5 | 66 | 61 |
| 0 | 0 | 0 | 61 | 2 | 7 |
| 0 | 0 | 61 | 12 | 71 | 61 |
| 51 | 69 | 0 | 0 | 0 | 0 |
| 30 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 68 | 7 | 5 |
| 0 | 0 | 68 | 7 | 5 | 0 |
| 0 | 0 | 12 | 0 | 0 | 5 |
| 0 | 0 | 61 | 12 | 71 | 61 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,72,1,71,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[22,7,0,0,0,0,4,51,0,0,0,0,0,0,5,66,0,61,0,0,68,5,61,12,0,0,7,66,2,71,0,0,5,61,7,61],[51,30,0,0,0,0,69,22,0,0,0,0,0,0,5,68,12,61,0,0,68,7,0,12,0,0,7,5,0,71,0,0,5,0,5,61] >;
C3×C23.36D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{36}D_4 % in TeX
G:=Group("C3xC2^3.36D4"); // GroupNames label
G:=SmallGroup(192,850);
// by ID
G=gap.SmallGroup(192,850);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations