direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.26D6, C24.79D6, (C22×C12)⋊16C4, (C23×C4).18S3, C6.43(C23×C4), C6⋊4(C42⋊C2), (C23×C12).17C2, (C2×C6).284C24, C4⋊Dic3⋊83C22, (C22×C4)⋊12Dic3, (C22×C4).464D6, C2.5(C23×Dic3), C12.181(C22×C4), (C2×C12).885C23, (C4×Dic3)⋊81C22, C22.41(S3×C23), C23.41(C2×Dic3), C4.39(C22×Dic3), C22.80(C4○D12), C23.242(C22×S3), (C23×C6).106C22, (C22×C6).413C23, (C22×C12).547C22, (C2×Dic3).278C23, C22.31(C22×Dic3), C6.D4.144C22, (C22×Dic3).229C22, (C2×C12)⋊37(C2×C4), C3⋊5(C2×C42⋊C2), (C2×C4×Dic3)⋊37C2, C6.60(C2×C4○D4), C2.5(C2×C4○D12), (C2×C4⋊Dic3)⋊50C2, (C2×C4)⋊11(C2×Dic3), (C2×C6).111(C4○D4), (C2×C4).829(C22×S3), (C2×C6).208(C22×C4), (C22×C6).143(C2×C4), (C2×C6.D4).25C2, SmallGroup(192,1345)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 568 in 330 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×28], C2×C4 [×16], C23, C23 [×6], C23 [×4], Dic3 [×8], C12 [×8], C2×C6, C2×C6 [×10], C2×C6 [×12], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×28], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C4×Dic3 [×8], C4⋊Dic3 [×8], C6.D4 [×8], C22×Dic3 [×4], C22×C12 [×2], C22×C12 [×12], C23×C6, C2×C42⋊C2, C2×C4×Dic3 [×2], C2×C4⋊Dic3 [×2], C23.26D6 [×8], C2×C6.D4 [×2], C23×C12, C2×C23.26D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C4○D4 [×4], C24, C2×Dic3 [×28], C22×S3 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C4○D12 [×4], C22×Dic3 [×14], S3×C23, C2×C42⋊C2, C23.26D6 [×4], C2×C4○D12 [×2], C23×Dic3, C2×C23.26D6
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=e5 >
►On 96 points
Generators in S96
(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 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 61)(21 62)(22 63)(23 64)(24 65)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 85)(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 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 58)(26 59)(27 60)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 85)(48 86)(61 83)(62 84)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 73)(23 74)(24 75)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)(49 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)
(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 87 67 59)(2 92 68 52)(3 85 69 57)(4 90 70 50)(5 95 71 55)(6 88 72 60)(7 93 61 53)(8 86 62 58)(9 91 63 51)(10 96 64 56)(11 89 65 49)(12 94 66 54)(13 44 82 33)(14 37 83 26)(15 42 84 31)(16 47 73 36)(17 40 74 29)(18 45 75 34)(19 38 76 27)(20 43 77 32)(21 48 78 25)(22 41 79 30)(23 46 80 35)(24 39 81 28)
G:=sub<Sym(96)| (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,61)(21,62)(22,63)(23,64)(24,65)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,85)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86)(61,83)(62,84)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94), (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), (1,87,67,59)(2,92,68,52)(3,85,69,57)(4,90,70,50)(5,95,71,55)(6,88,72,60)(7,93,61,53)(8,86,62,58)(9,91,63,51)(10,96,64,56)(11,89,65,49)(12,94,66,54)(13,44,82,33)(14,37,83,26)(15,42,84,31)(16,47,73,36)(17,40,74,29)(18,45,75,34)(19,38,76,27)(20,43,77,32)(21,48,78,25)(22,41,79,30)(23,46,80,35)(24,39,81,28)>;
G:=Group( (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,61)(21,62)(22,63)(23,64)(24,65)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,85)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86)(61,83)(62,84)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94), (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), (1,87,67,59)(2,92,68,52)(3,85,69,57)(4,90,70,50)(5,95,71,55)(6,88,72,60)(7,93,61,53)(8,86,62,58)(9,91,63,51)(10,96,64,56)(11,89,65,49)(12,94,66,54)(13,44,82,33)(14,37,83,26)(15,42,84,31)(16,47,73,36)(17,40,74,29)(18,45,75,34)(19,38,76,27)(20,43,77,32)(21,48,78,25)(22,41,79,30)(23,46,80,35)(24,39,81,28) );
G=PermutationGroup([(1,83),(2,84),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,61),(21,62),(22,63),(23,64),(24,65),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,85),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,58),(26,59),(27,60),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,85),(48,86),(61,83),(62,84),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(71,81),(72,82)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,73),(23,74),(24,75),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41),(49,95),(50,96),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94)], [(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)], [(1,87,67,59),(2,92,68,52),(3,85,69,57),(4,90,70,50),(5,95,71,55),(6,88,72,60),(7,93,61,53),(8,86,62,58),(9,91,63,51),(10,96,64,56),(11,89,65,49),(12,94,66,54),(13,44,82,33),(14,37,83,26),(15,42,84,31),(16,47,73,36),(17,40,74,29),(18,45,75,34),(19,38,76,27),(20,43,77,32),(21,48,78,25),(22,41,79,30),(23,46,80,35),(24,39,81,28)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 7 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 3 | 10 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,2,1],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,11,0,0,0,9,7],[8,0,0,0,0,1,0,0,0,0,3,3,0,0,6,10] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4AB | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C2×C23.26D6 | C2×C4×Dic3 | C2×C4⋊Dic3 | C23.26D6 | C2×C6.D4 | C23×C12 | C22×C12 | C23×C4 | C22×C4 | C22×C4 | C24 | C2×C6 | C22 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 16 | 1 | 8 | 6 | 1 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{26}D_6 % in TeX
G:=Group("C2xC2^3.26D6"); // GroupNames label
G:=SmallGroup(192,1345);
// by ID
G=gap.SmallGroup(192,1345);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,1123,6278]);
// 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=e^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀