direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.28D6, C24.81D6, (C23×C4)⋊8S3, (C23×C12)⋊4C2, (C22×C4)⋊46D6, D6⋊C4⋊41C22, (C2×C6).287C24, C6.133(C22×D4), (C22×C6).205D4, (C2×C12).704C23, Dic3⋊C4⋊44C22, (C22×C12)⋊56C22, C6⋊4(C22.D4), C23.98(C3⋊D4), C22.82(C4○D12), C6.D4⋊55C22, (S3×C23).74C22, (C22×C6).416C23, C23.243(C22×S3), C22.302(S3×C23), (C23×C6).109C22, (C22×S3).125C23, (C2×Dic3).149C23, (C22×Dic3).161C22, (C2×D6⋊C4)⋊13C2, C6.62(C2×C4○D4), C2.70(C2×C4○D12), (C2×C6).574(C2×D4), C2.6(C22×C3⋊D4), C3⋊5(C2×C22.D4), (C2×Dic3⋊C4)⋊18C2, (C2×C6).113(C4○D4), (C2×C6.D4)⋊22C2, (C2×C4).657(C22×S3), (C22×C3⋊D4).13C2, C22.103(C2×C3⋊D4), (C2×C3⋊D4).136C22, SmallGroup(192,1348)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.28D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 824 in 342 conjugacy classes, 127 normal (17 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, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, S3×C23, C23×C6, C2×C22.D4, C2×Dic3⋊C4, C2×D6⋊C4, C23.28D6, C2×C6.D4, C22×C3⋊D4, C23×C12, C2×C23.28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22.D4, C22×D4, C2×C4○D4, C4○D12, C2×C3⋊D4, S3×C23, C2×C22.D4, C23.28D6, C2×C4○D12, C22×C3⋊D4, C2×C23.28D6
►On 96 points
Generators in S96
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 95)(14 96)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)
(1 95)(2 96)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 56)(14 57)(15 58)(16 59)(17 60)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 61)(36 62)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 73)(45 74)(46 75)(47 76)(48 77)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 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 6 89 94)(2 93 90 5)(3 4 91 92)(7 12 95 88)(8 87 96 11)(9 10 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 72 69 28)(26 27 70 71)(29 68 61 36)(30 35 62 67)(31 66 63 34)(32 33 64 65)(37 40 84 75)(38 74 73 39)(41 48 76 83)(42 82 77 47)(43 46 78 81)(44 80 79 45)
G:=sub<Sym(96)| (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,56)(2,57)(3,58)(4,59)(5,60)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,95)(14,96)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84), (1,95)(2,96)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,6,89,94)(2,93,90,5)(3,4,91,92)(7,12,95,88)(8,87,96,11)(9,10,85,86)(13,24,50,49)(14,60,51,23)(15,22,52,59)(16,58,53,21)(17,20,54,57)(18,56,55,19)(25,72,69,28)(26,27,70,71)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(37,40,84,75)(38,74,73,39)(41,48,76,83)(42,82,77,47)(43,46,78,81)(44,80,79,45)>;
G:=Group( (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,56)(2,57)(3,58)(4,59)(5,60)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,95)(14,96)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84), (1,95)(2,96)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,6,89,94)(2,93,90,5)(3,4,91,92)(7,12,95,88)(8,87,96,11)(9,10,85,86)(13,24,50,49)(14,60,51,23)(15,22,52,59)(16,58,53,21)(17,20,54,57)(18,56,55,19)(25,72,69,28)(26,27,70,71)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(37,40,84,75)(38,74,73,39)(41,48,76,83)(42,82,77,47)(43,46,78,81)(44,80,79,45) );
G=PermutationGroup([[(1,36),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(37,60),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(61,94),(62,95),(63,96),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,95),(14,96),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84)], [(1,95),(2,96),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,56),(14,57),(15,58),(16,59),(17,60),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,61),(36,62),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,73),(45,74),(46,75),(47,76),(48,77)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(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)], [(1,6,89,94),(2,93,90,5),(3,4,91,92),(7,12,95,88),(8,87,96,11),(9,10,85,86),(13,24,50,49),(14,60,51,23),(15,22,52,59),(16,58,53,21),(17,20,54,57),(18,56,55,19),(25,72,69,28),(26,27,70,71),(29,68,61,36),(30,35,62,67),(31,66,63,34),(32,33,64,65),(37,40,84,75),(38,74,73,39),(41,48,76,83),(42,82,77,47),(43,46,78,81),(44,80,79,45)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | ··· | 4H | 4I | ··· | 4N | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 |
| kernel | C2×C23.28D6 | C2×Dic3⋊C4 | C2×D6⋊C4 | C23.28D6 | C2×C6.D4 | C22×C3⋊D4 | C23×C12 | C23×C4 | C22×C6 | C22×C4 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 | 6 | 1 | 8 | 8 | 16 |
Matrix representation of C2×C23.28D6 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 | 8 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,10,5],[1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,5,0,0,0,0,0,3,8] >;
C2×C23.28D6 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{28}D_6 % in TeX
G:=Group("C2xC2^3.28D6"); // GroupNames label
G:=SmallGroup(192,1348);
// by ID
G=gap.SmallGroup(192,1348);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations