direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4○D12, C6.4C25, D6.1C24, C24.84D6, D12⋊14C23, C12.77C24, Dic6⋊13C23, Dic3.2C24, (C4×S3)⋊8C23, (C23×C4)⋊12S3, (C22×C4)⋊49D6, C3⋊D4⋊7C23, C2.5(S3×C24), (C23×C12)⋊11C2, (C2×C12)⋊15C23, C4.76(S3×C23), (C2×D12)⋊66C22, (C22×D12)⋊25C2, (C2×C6).326C24, C22.7(S3×C23), (C22×C12)⋊62C22, (C22×Dic6)⋊26C2, (C2×Dic6)⋊77C22, (C23×C6).116C22, C23.357(C22×S3), (C22×C6).433C23, (C22×S3).245C23, (S3×C23).116C22, (C2×Dic3).296C23, (C22×Dic3).239C22, C6⋊1(C2×C4○D4), C3⋊1(C22×C4○D4), (S3×C2×C4)⋊72C22, (S3×C22×C4)⋊26C2, (C2×C6)⋊13(C4○D4), (C2×C4)⋊12(C22×S3), (C22×C3⋊D4)⋊22C2, (C2×C3⋊D4)⋊56C22, SmallGroup(192,1513)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1784 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×12], C3, C4 [×8], C4 [×8], C22 [×11], C22 [×44], S3 [×8], C6, C6 [×6], C6 [×4], C2×C4 [×28], C2×C4 [×44], D4 [×48], Q8 [×16], C23, C23 [×6], C23 [×24], Dic3 [×8], C12 [×8], D6 [×8], D6 [×24], C2×C6 [×11], C2×C6 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24, C24 [×2], Dic6 [×16], C4×S3 [×32], D12 [×16], C2×Dic3 [×12], C3⋊D4 [×32], C2×C12 [×28], C22×S3 [×12], C22×S3 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C23×C4, C23×C4 [×2], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C2×Dic6 [×12], S3×C2×C4 [×24], C2×D12 [×12], C4○D12 [×64], C22×Dic3 [×2], C2×C3⋊D4 [×24], C22×C12 [×2], C22×C12 [×12], S3×C23 [×2], C23×C6, C22×C4○D4, C22×Dic6, S3×C22×C4 [×2], C22×D12, C2×C4○D12 [×24], C22×C3⋊D4 [×2], C23×C12, C22×C4○D12
Quotients:
C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C4○D4 [×4], C24 [×31], C22×S3 [×35], C2×C4○D4 [×6], C25, C4○D12 [×4], S3×C23 [×15], C22×C4○D4, C2×C4○D12 [×6], S3×C24, C22×C4○D12
Generators and relations
G = < a,b,c,d,e | a2=b2=c4=e2=1, d6=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >
►On 96 points
(1 84)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 85)(71 86)(72 87)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 85)(11 86)(12 87)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 57)(26 58)(27 59)(28 60)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(61 84)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)
(1 48 7 42)(2 37 8 43)(3 38 9 44)(4 39 10 45)(5 40 11 46)(6 41 12 47)(13 96 19 90)(14 85 20 91)(15 86 21 92)(16 87 22 93)(17 88 23 94)(18 89 24 95)(25 74 31 80)(26 75 32 81)(27 76 33 82)(28 77 34 83)(29 78 35 84)(30 79 36 73)(49 67 55 61)(50 68 56 62)(51 69 57 63)(52 70 58 64)(53 71 59 65)(54 72 60 66)
(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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 26)(14 25)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(48 60)(73 86)(74 85)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)
G:=sub<Sym(96)| (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,85)(71,86)(72,87), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,57)(26,58)(27,59)(28,60)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (1,48,7,42)(2,37,8,43)(3,38,9,44)(4,39,10,45)(5,40,11,46)(6,41,12,47)(13,96,19,90)(14,85,20,91)(15,86,21,92)(16,87,22,93)(17,88,23,94)(18,89,24,95)(25,74,31,80)(26,75,32,81)(27,76,33,82)(28,77,34,83)(29,78,35,84)(30,79,36,73)(49,67,55,61)(50,68,56,62)(51,69,57,63)(52,70,58,64)(53,71,59,65)(54,72,60,66), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,26)(14,25)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(73,86)(74,85)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)>;
G:=Group( (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,85)(71,86)(72,87), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,37)(25,57)(26,58)(27,59)(28,60)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(61,84)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83), (1,48,7,42)(2,37,8,43)(3,38,9,44)(4,39,10,45)(5,40,11,46)(6,41,12,47)(13,96,19,90)(14,85,20,91)(15,86,21,92)(16,87,22,93)(17,88,23,94)(18,89,24,95)(25,74,31,80)(26,75,32,81)(27,76,33,82)(28,77,34,83)(29,78,35,84)(30,79,36,73)(49,67,55,61)(50,68,56,62)(51,69,57,63)(52,70,58,64)(53,71,59,65)(54,72,60,66), (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,26)(14,25)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(73,86)(74,85)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87) );
G=PermutationGroup([(1,84),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,44),(26,45),(27,46),(28,47),(29,48),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(36,43),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,85),(71,86),(72,87)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,85),(11,86),(12,87),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,37),(25,57),(26,58),(27,59),(28,60),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(61,84),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83)], [(1,48,7,42),(2,37,8,43),(3,38,9,44),(4,39,10,45),(5,40,11,46),(6,41,12,47),(13,96,19,90),(14,85,20,91),(15,86,21,92),(16,87,22,93),(17,88,23,94),(18,89,24,95),(25,74,31,80),(26,75,32,81),(27,76,33,82),(28,77,34,83),(29,78,35,84),(30,79,36,73),(49,67,55,61),(50,68,56,62),(51,69,57,63),(52,70,58,64),(53,71,59,65),(54,72,60,66)], [(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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,26),(14,25),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(48,60),(73,86),(74,85),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,5,0] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 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 | 6 | ··· | 6 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C22×C4○D12 | C22×Dic6 | S3×C22×C4 | C22×D12 | C2×C4○D12 | C22×C3⋊D4 | C23×C12 | C23×C4 | C22×C4 | C24 | C2×C6 | C22 |
| # reps | 1 | 1 | 2 | 1 | 24 | 2 | 1 | 1 | 14 | 1 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_{12} % in TeX
G:=Group("C2^2xC4oD12"); // GroupNames label
G:=SmallGroup(192,1513);
// by ID
G=gap.SmallGroup(192,1513);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^6=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations