metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊3D12, C24.26D6, (C2×C12)⋊6D4, (C22×C6)⋊7D4, (C2×Dic3)⋊5D4, (C22×S3)⋊4D4, C6.33C22≀C2, (C22×D12)⋊3C2, C3⋊2(C23⋊2D4), (C22×C4).51D6, C2.8(C23⋊2D6), C6.59(C4⋊D4), C2.7(C12⋊7D4), C2.6(C12⋊3D4), C6.13(C4⋊1D4), C22.242(S3×D4), C2.34(D6⋊D4), C2.34(Dic3⋊D4), C6.C42⋊32C2, C22.126(C2×D12), (C23×C6).43C22, (S3×C23).16C22, (C22×C12).61C22, (C22×C6).334C23, C23.382(C22×S3), C22.100(C4○D12), (C22×Dic3).46C22, (C2×D6⋊C4)⋊8C2, (C2×C4)⋊3(C3⋊D4), (C2×C22⋊C4)⋊8S3, (C6×C22⋊C4)⋊11C2, (C2×C6).325(C2×D4), (C22×C3⋊D4)⋊1C2, (C2×C6).80(C4○D4), C22.128(C2×C3⋊D4), SmallGroup(192,519)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊3D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1080 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C22×D4, D6⋊C4, C3×C22⋊C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C23⋊2D4, C6.C42, C2×D6⋊C4, C6×C22⋊C4, C22×D12, C22×C3⋊D4, C23⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C22≀C2, C4⋊D4, C4⋊1D4, C2×D12, C4○D12, S3×D4, C2×C3⋊D4, C23⋊2D4, D6⋊D4, Dic3⋊D4, C12⋊7D4, C23⋊2D6, C12⋊3D4, C23⋊3D12
►On 96 points
Generators in S96
(1 91)(2 15)(3 93)(4 17)(5 95)(6 19)(7 85)(8 21)(9 87)(10 23)(11 89)(12 13)(14 43)(16 45)(18 47)(20 37)(22 39)(24 41)(25 69)(26 58)(27 71)(28 60)(29 61)(30 50)(31 63)(32 52)(33 65)(34 54)(35 67)(36 56)(38 86)(40 88)(42 90)(44 92)(46 94)(48 96)(49 78)(51 80)(53 82)(55 84)(57 74)(59 76)(62 79)(64 81)(66 83)(68 73)(70 75)(72 77)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 56)(14 57)(15 58)(16 59)(17 60)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(61 95)(62 96)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 73)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(61 86)(62 85)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)
G:=sub<Sym(96)| (1,91)(2,15)(3,93)(4,17)(5,95)(6,19)(7,85)(8,21)(9,87)(10,23)(11,89)(12,13)(14,43)(16,45)(18,47)(20,37)(22,39)(24,41)(25,69)(26,58)(27,71)(28,60)(29,61)(30,50)(31,63)(32,52)(33,65)(34,54)(35,67)(36,56)(38,86)(40,88)(42,90)(44,92)(46,94)(48,96)(49,78)(51,80)(53,82)(55,84)(57,74)(59,76)(62,79)(64,81)(66,83)(68,73)(70,75)(72,77), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(61,95)(62,96)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(61,86)(62,85)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80)>;
G:=Group( (1,91)(2,15)(3,93)(4,17)(5,95)(6,19)(7,85)(8,21)(9,87)(10,23)(11,89)(12,13)(14,43)(16,45)(18,47)(20,37)(22,39)(24,41)(25,69)(26,58)(27,71)(28,60)(29,61)(30,50)(31,63)(32,52)(33,65)(34,54)(35,67)(36,56)(38,86)(40,88)(42,90)(44,92)(46,94)(48,96)(49,78)(51,80)(53,82)(55,84)(57,74)(59,76)(62,79)(64,81)(66,83)(68,73)(70,75)(72,77), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(61,95)(62,96)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,73)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(61,86)(62,85)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80) );
G=PermutationGroup([[(1,91),(2,15),(3,93),(4,17),(5,95),(6,19),(7,85),(8,21),(9,87),(10,23),(11,89),(12,13),(14,43),(16,45),(18,47),(20,37),(22,39),(24,41),(25,69),(26,58),(27,71),(28,60),(29,61),(30,50),(31,63),(32,52),(33,65),(34,54),(35,67),(36,56),(38,86),(40,88),(42,90),(44,92),(46,94),(48,96),(49,78),(51,80),(53,82),(55,84),(57,74),(59,76),(62,79),(64,81),(66,83),(68,73),(70,75),(72,77)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,56),(14,57),(15,58),(16,59),(17,60),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(37,80),(38,81),(39,82),(40,83),(41,84),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(61,95),(62,96),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,73),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,60),(23,59),(24,58),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(61,86),(62,85),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,74),(75,84),(76,83),(77,82),(78,81),(79,80)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D12 | S3×D4 |
| kernel | C23⋊3D12 | C6.C42 | C2×D6⋊C4 | C6×C22⋊C4 | C22×D12 | C22×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of C23⋊3D12 ►in GL6(𝔽13)
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[8,8,0,0,0,0,10,5,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C23⋊3D12 in GAP, Magma, Sage, TeX
C_2^3\rtimes_3D_{12} % in TeX
G:=Group("C2^3:3D12"); // GroupNames label
G:=SmallGroup(192,519);
// by ID
G=gap.SmallGroup(192,519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations