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
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
Generators and relations
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 >
►On 96 points
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(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 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 61)(34 62)(35 63)(36 64)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)
(1 32)(2 87)(3 34)(4 89)(5 36)(6 91)(7 26)(8 93)(9 28)(10 95)(11 30)(12 85)(13 59)(14 47)(15 49)(16 37)(17 51)(18 39)(19 53)(20 41)(21 55)(22 43)(23 57)(24 45)(25 76)(27 78)(29 80)(31 82)(33 84)(35 74)(38 68)(40 70)(42 72)(44 62)(46 64)(48 66)(50 67)(52 69)(54 71)(56 61)(58 63)(60 65)(73 88)(75 90)(77 92)(79 94)(81 96)(83 86)
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 49)
(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 82 83 12)(2 11 84 81)(3 80 73 10)(4 9 74 79)(5 78 75 8)(6 7 76 77)(13 16 64 67)(14 66 65 15)(17 24 68 63)(18 62 69 23)(19 22 70 61)(20 72 71 21)(25 92 91 26)(27 90 93 36)(28 35 94 89)(29 88 95 34)(30 33 96 87)(31 86 85 32)(37 46 50 59)(38 58 51 45)(39 44 52 57)(40 56 53 43)(41 42 54 55)(47 48 60 49)
G:=sub<Sym(96)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76), (1,32)(2,87)(3,34)(4,89)(5,36)(6,91)(7,26)(8,93)(9,28)(10,95)(11,30)(12,85)(13,59)(14,47)(15,49)(16,37)(17,51)(18,39)(19,53)(20,41)(21,55)(22,43)(23,57)(24,45)(25,76)(27,78)(29,80)(31,82)(33,84)(35,74)(38,68)(40,70)(42,72)(44,62)(46,64)(48,66)(50,67)(52,69)(54,71)(56,61)(58,63)(60,65)(73,88)(75,90)(77,92)(79,94)(81,96)(83,86), (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49), (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), (1,82,83,12)(2,11,84,81)(3,80,73,10)(4,9,74,79)(5,78,75,8)(6,7,76,77)(13,16,64,67)(14,66,65,15)(17,24,68,63)(18,62,69,23)(19,22,70,61)(20,72,71,21)(25,92,91,26)(27,90,93,36)(28,35,94,89)(29,88,95,34)(30,33,96,87)(31,86,85,32)(37,46,50,59)(38,58,51,45)(39,44,52,57)(40,56,53,43)(41,42,54,55)(47,48,60,49)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,85)(21,86)(22,87)(23,88)(24,89)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,73)(58,74)(59,75)(60,76), (1,32)(2,87)(3,34)(4,89)(5,36)(6,91)(7,26)(8,93)(9,28)(10,95)(11,30)(12,85)(13,59)(14,47)(15,49)(16,37)(17,51)(18,39)(19,53)(20,41)(21,55)(22,43)(23,57)(24,45)(25,76)(27,78)(29,80)(31,82)(33,84)(35,74)(38,68)(40,70)(42,72)(44,62)(46,64)(48,66)(50,67)(52,69)(54,71)(56,61)(58,63)(60,65)(73,88)(75,90)(77,92)(79,94)(81,96)(83,86), (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49), (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), (1,82,83,12)(2,11,84,81)(3,80,73,10)(4,9,74,79)(5,78,75,8)(6,7,76,77)(13,16,64,67)(14,66,65,15)(17,24,68,63)(18,62,69,23)(19,22,70,61)(20,72,71,21)(25,92,91,26)(27,90,93,36)(28,35,94,89)(29,88,95,34)(30,33,96,87)(31,86,85,32)(37,46,50,59)(38,58,51,45)(39,44,52,57)(40,56,53,43)(41,42,54,55)(47,48,60,49) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,37),(9,38),(10,39),(11,40),(12,41),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,85),(21,86),(22,87),(23,88),(24,89),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,61),(34,62),(35,63),(36,64),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,73),(58,74),(59,75),(60,76)], [(1,32),(2,87),(3,34),(4,89),(5,36),(6,91),(7,26),(8,93),(9,28),(10,95),(11,30),(12,85),(13,59),(14,47),(15,49),(16,37),(17,51),(18,39),(19,53),(20,41),(21,55),(22,43),(23,57),(24,45),(25,76),(27,78),(29,80),(31,82),(33,84),(35,74),(38,68),(40,70),(42,72),(44,62),(46,64),(48,66),(50,67),(52,69),(54,71),(56,61),(58,63),(60,65),(73,88),(75,90),(77,92),(79,94),(81,96),(83,86)], [(1,83),(2,84),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,61),(23,62),(24,63),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,49)], [(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)], [(1,82,83,12),(2,11,84,81),(3,80,73,10),(4,9,74,79),(5,78,75,8),(6,7,76,77),(13,16,64,67),(14,66,65,15),(17,24,68,63),(18,62,69,23),(19,22,70,61),(20,72,71,21),(25,92,91,26),(27,90,93,36),(28,35,94,89),(29,88,95,34),(30,33,96,87),(31,86,85,32),(37,46,50,59),(38,58,51,45),(39,44,52,57),(40,56,53,43),(41,42,54,55),(47,48,60,49)])
Matrix representation ►G ⊆ 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] >;
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 |
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