direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.D12, C24.66D6, C23.56D12, C22⋊C4⋊42D6, C6.7(C22×D4), D6⋊C4⋊48C22, (C2×C6).36C24, C2.9(C22×D12), C4⋊Dic3⋊52C22, (C23×Dic3)⋊4C2, (C22×C6).117D4, (C22×C4).187D6, C22.66(C2×D12), (C2×C12).129C23, C6⋊2(C22.D4), (C22×S3).8C23, C22.75(S3×C23), (C23×C6).62C22, (C2×Dic3).9C23, (S3×C23).33C22, C23.157(C22×S3), (C22×C12).72C22, (C22×C6).126C23, C22.69(D4⋊2S3), (C22×Dic3)⋊42C22, (C2×D6⋊C4)⋊19C2, C6.68(C2×C4○D4), (C2×C6).48(C2×D4), (C2×C22⋊C4)⋊15S3, (C6×C22⋊C4)⋊14C2, (C2×C4⋊Dic3)⋊20C2, C3⋊2(C2×C22.D4), C2.11(C2×D4⋊2S3), (C2×C6).168(C4○D4), (C3×C22⋊C4)⋊47C22, (C2×C4).135(C22×S3), (C2×C3⋊D4).91C22, (C22×C3⋊D4).12C2, SmallGroup(192,1051)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.D12
G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 872 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×10], C22, C22 [×10], C22 [×22], S3 [×2], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×6], C23 [×12], Dic3 [×6], C12 [×4], D6 [×10], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×11], C2×D4 [×8], C24, C24, C2×Dic3 [×6], C2×Dic3 [×14], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×4], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C4⋊Dic3 [×8], D6⋊C4 [×8], C3×C22⋊C4 [×4], C22×Dic3, C22×Dic3 [×6], C22×Dic3 [×4], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C2×C22.D4, C22.D12 [×8], C2×C4⋊Dic3 [×2], C2×D6⋊C4 [×2], C6×C22⋊C4, C23×Dic3, C22×C3⋊D4, C2×C22.D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, D12 [×4], C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×D12 [×6], D4⋊2S3 [×4], S3×C23, C2×C22.D4, C22.D12 [×4], C22×D12, C2×D4⋊2S3 [×2], C2×C22.D12
►On 96 points
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 61)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 92)(26 93)(27 94)(28 95)(29 96)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)
(1 13)(2 51)(3 15)(4 53)(5 17)(6 55)(7 19)(8 57)(9 21)(10 59)(11 23)(12 49)(14 93)(16 95)(18 85)(20 87)(22 89)(24 91)(25 40)(26 75)(27 42)(28 77)(29 44)(30 79)(31 46)(32 81)(33 48)(34 83)(35 38)(36 73)(37 71)(39 61)(41 63)(43 65)(45 67)(47 69)(50 92)(52 94)(54 96)(56 86)(58 88)(60 90)(62 74)(64 76)(66 78)(68 80)(70 82)(72 84)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 83)(38 84)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)
(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 91 92 12)(2 11 93 90)(3 89 94 10)(4 9 95 88)(5 87 96 8)(6 7 85 86)(13 24 50 49)(14 60 51 23)(15 22 52 59)(16 58 53 21)(17 20 54 57)(18 56 55 19)(25 61 62 36)(26 35 63 72)(27 71 64 34)(28 33 65 70)(29 69 66 32)(30 31 67 68)(37 76 83 42)(38 41 84 75)(39 74 73 40)(43 82 77 48)(44 47 78 81)(45 80 79 46)
G:=sub<Sym(96)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (1,13)(2,51)(3,15)(4,53)(5,17)(6,55)(7,19)(8,57)(9,21)(10,59)(11,23)(12,49)(14,93)(16,95)(18,85)(20,87)(22,89)(24,91)(25,40)(26,75)(27,42)(28,77)(29,44)(30,79)(31,46)(32,81)(33,48)(34,83)(35,38)(36,73)(37,71)(39,61)(41,63)(43,65)(45,67)(47,69)(50,92)(52,94)(54,96)(56,86)(58,88)(60,90)(62,74)(64,76)(66,78)(68,80)(70,82)(72,84), (1,92)(2,93)(3,94)(4,95)(5,96)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,83)(38,84)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82), (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), (1,91,92,12)(2,11,93,90)(3,89,94,10)(4,9,95,88)(5,87,96,8)(6,7,85,86)(13,24,50,49)(14,60,51,23)(15,22,52,59)(16,58,53,21)(17,20,54,57)(18,56,55,19)(25,61,62,36)(26,35,63,72)(27,71,64,34)(28,33,65,70)(29,69,66,32)(30,31,67,68)(37,76,83,42)(38,41,84,75)(39,74,73,40)(43,82,77,48)(44,47,78,81)(45,80,79,46)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (1,13)(2,51)(3,15)(4,53)(5,17)(6,55)(7,19)(8,57)(9,21)(10,59)(11,23)(12,49)(14,93)(16,95)(18,85)(20,87)(22,89)(24,91)(25,40)(26,75)(27,42)(28,77)(29,44)(30,79)(31,46)(32,81)(33,48)(34,83)(35,38)(36,73)(37,71)(39,61)(41,63)(43,65)(45,67)(47,69)(50,92)(52,94)(54,96)(56,86)(58,88)(60,90)(62,74)(64,76)(66,78)(68,80)(70,82)(72,84), (1,92)(2,93)(3,94)(4,95)(5,96)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,83)(38,84)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82), (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), (1,91,92,12)(2,11,93,90)(3,89,94,10)(4,9,95,88)(5,87,96,8)(6,7,85,86)(13,24,50,49)(14,60,51,23)(15,22,52,59)(16,58,53,21)(17,20,54,57)(18,56,55,19)(25,61,62,36)(26,35,63,72)(27,71,64,34)(28,33,65,70)(29,69,66,32)(30,31,67,68)(37,76,83,42)(38,41,84,75)(39,74,73,40)(43,82,77,48)(44,47,78,81)(45,80,79,46) );
G=PermutationGroup([(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,61),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,92),(26,93),(27,94),(28,95),(29,96),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58)], [(1,13),(2,51),(3,15),(4,53),(5,17),(6,55),(7,19),(8,57),(9,21),(10,59),(11,23),(12,49),(14,93),(16,95),(18,85),(20,87),(22,89),(24,91),(25,40),(26,75),(27,42),(28,77),(29,44),(30,79),(31,46),(32,81),(33,48),(34,83),(35,38),(36,73),(37,71),(39,61),(41,63),(43,65),(45,67),(47,69),(50,92),(52,94),(54,96),(56,86),(58,88),(60,90),(62,74),(64,76),(66,78),(68,80),(70,82),(72,84)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,61),(37,83),(38,84),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82)], [(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)], [(1,91,92,12),(2,11,93,90),(3,89,94,10),(4,9,95,88),(5,87,96,8),(6,7,85,86),(13,24,50,49),(14,60,51,23),(15,22,52,59),(16,58,53,21),(17,20,54,57),(18,56,55,19),(25,61,62,36),(26,35,63,72),(27,71,64,34),(28,33,65,70),(29,69,66,32),(30,31,67,68),(37,76,83,42),(38,41,84,75),(39,74,73,40),(43,82,77,48),(44,47,78,81),(45,80,79,46)])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 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 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | D4⋊2S3 |
| kernel | C2×C22.D12 | C22.D12 | C2×C4⋊Dic3 | C2×D6⋊C4 | C6×C22⋊C4 | C23×Dic3 | C22×C3⋊D4 | C2×C22⋊C4 | C22×C6 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 8 | 8 | 4 |
Matrix representation of C2×C22.D12 ►in GL6(𝔽13)
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 3 | 6 |
G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[5,3,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;
C2×C22.D12 in GAP, Magma, Sage, TeX
C_2\times C_2^2.D_{12} % in TeX
G:=Group("C2xC2^2.D12"); // GroupNames label
G:=SmallGroup(192,1051);
// by ID
G=gap.SmallGroup(192,1051);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations