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, (C22×C6).205D4, C6.133(C22×D4), (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, (C23×C6).109C22, C22.302(S3×C23), (C22×C6).416C23, C23.243(C22×S3), (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), C3⋊5(C2×C22.D4), C2.6(C22×C3⋊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
Subgroups: 824 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 [×12], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×7], C2×D4 [×8], C24, C24, C2×Dic3 [×6], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×12], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×8], D6⋊C4 [×8], C6.D4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×6], C22×C12 [×4], S3×C23, C23×C6, C2×C22.D4, C2×Dic3⋊C4 [×2], C2×D6⋊C4 [×2], C23.28D6 [×8], C2×C6.D4, C22×C3⋊D4, C23×C12, C2×C23.28D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C4○D12 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22.D4, C23.28D6 [×4], C2×C4○D12 [×2], C22×C3⋊D4, C2×C23.28D6
Generators and relations
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 >
►On 96 points
Generators in S96
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(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 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 49)(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 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 61)(34 62)(35 63)(36 64)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(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 70 71 36)(26 35 72 69)(27 68 61 34)(28 33 62 67)(29 66 63 32)(30 31 64 65)(37 48 74 73)(38 84 75 47)(39 46 76 83)(40 82 77 45)(41 44 78 81)(42 80 79 43)
G:=sub<Sym(96)| (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(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)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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)(25,70,71,36)(26,35,72,69)(27,68,61,34)(28,33,62,67)(29,66,63,32)(30,31,64,65)(37,48,74,73)(38,84,75,47)(39,46,76,83)(40,82,77,45)(41,44,78,81)(42,80,79,43)>;
G:=Group( (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(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)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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)(25,70,71,36)(26,35,72,69)(27,68,61,34)(28,33,62,67)(29,66,63,32)(30,31,64,65)(37,48,74,73)(38,84,75,47)(39,46,76,83)(40,82,77,45)(41,44,78,81)(42,80,79,43) );
G=PermutationGroup([(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,49),(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),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,61),(34,62),(35,63),(36,64),(37,80),(38,81),(39,82),(40,83),(41,84),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79)], [(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),(25,70,71,36),(26,35,72,69),(27,68,61,34),(28,33,62,67),(29,66,63,32),(30,31,64,65),(37,48,74,73),(38,84,75,47),(39,46,76,83),(40,82,77,45),(41,44,78,81),(42,80,79,43)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀