metabelian, supersoluble, monomial
Aliases: C62.72C23, D6⋊3(C4×S3), C32⋊9(C4×D4), D6⋊C4⋊10S3, C3⋊Dic3⋊8D4, C6.40(S3×D4), D6⋊S3⋊6C4, (C2×C12).200D6, C2.3(D6⋊D6), (C2×Dic3).74D6, (C22×S3).43D6, C3⋊3(Dic3⋊4D4), C6.64(D4⋊2S3), (C6×C12).233C22, C62.C22⋊15C2, C2.3(D6.4D6), (C6×Dic3).67C22, C2.20(C4×S32), (C2×C4).96S32, C6.19(S3×C2×C4), (S3×C6)⋊6(C2×C4), (C3×D6⋊C4)⋊23C2, C3⋊Dic3⋊7(C2×C4), C22.40(C2×S32), (C2×S3×Dic3)⋊16C2, (C3×C6).56(C2×D4), (C4×C3⋊Dic3)⋊17C2, (S3×C2×C6).26C22, (C2×D6⋊S3).4C2, (C3×C6).69(C4○D4), (C3×C6).18(C22×C4), (C2×C6).91(C22×S3), (C2×C3⋊Dic3).134C22, SmallGroup(288,550)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.72C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=ece-1=b3c, ede-1=b3d >
Subgroups: 714 in 203 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, S3×Dic3, D6⋊S3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3⋊4D4, C62.C22, C3×D6⋊C4, C4×C3⋊Dic3, C2×S3×Dic3, C2×D6⋊S3, C62.72C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S32, S3×C2×C4, S3×D4, D4⋊2S3, C2×S32, Dic3⋊4D4, C4×S32, D6⋊D6, D6.4D6, C62.72C23
►On 96 points
Generators in S96
(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 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 69)(8 68)(9 67)(10 72)(11 71)(12 70)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(25 50)(26 49)(27 54)(28 53)(29 52)(30 51)(55 88)(56 87)(57 86)(58 85)(59 90)(60 89)(61 81)(62 80)(63 79)(64 84)(65 83)(66 82)(73 92)(74 91)(75 96)(76 95)(77 94)(78 93)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 86 10 89)(8 87 11 90)(9 88 12 85)(13 27 16 30)(14 28 17 25)(15 29 18 26)(31 52 34 49)(32 53 35 50)(33 54 36 51)(37 47 40 44)(38 48 41 45)(39 43 42 46)(55 73 58 76)(56 74 59 77)(57 75 60 78)(61 72 64 69)(62 67 65 70)(63 68 66 71)(79 94 82 91)(80 95 83 92)(81 96 84 93)
G:=sub<Sym(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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,95,11,93,9,91)(8,96,12,94,10,92)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39)(43,53,45,49,47,51)(44,54,46,50,48,52)(55,63,57,65,59,61)(56,64,58,66,60,62)(67,74,69,76,71,78)(68,75,70,77,72,73)(79,86,83,90,81,88)(80,87,84,85,82,89), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,69)(8,68)(9,67)(10,72)(11,71)(12,70)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,50)(26,49)(27,54)(28,53)(29,52)(30,51)(55,88)(56,87)(57,86)(58,85)(59,90)(60,89)(61,81)(62,80)(63,79)(64,84)(65,83)(66,82)(73,92)(74,91)(75,96)(76,95)(77,94)(78,93), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,86,10,89)(8,87,11,90)(9,88,12,85)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,52,34,49)(32,53,35,50)(33,54,36,51)(37,47,40,44)(38,48,41,45)(39,43,42,46)(55,73,58,76)(56,74,59,77)(57,75,60,78)(61,72,64,69)(62,67,65,70)(63,68,66,71)(79,94,82,91)(80,95,83,92)(81,96,84,93)>;
G:=Group( (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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,95,11,93,9,91)(8,96,12,94,10,92)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39)(43,53,45,49,47,51)(44,54,46,50,48,52)(55,63,57,65,59,61)(56,64,58,66,60,62)(67,74,69,76,71,78)(68,75,70,77,72,73)(79,86,83,90,81,88)(80,87,84,85,82,89), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,69)(8,68)(9,67)(10,72)(11,71)(12,70)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,50)(26,49)(27,54)(28,53)(29,52)(30,51)(55,88)(56,87)(57,86)(58,85)(59,90)(60,89)(61,81)(62,80)(63,79)(64,84)(65,83)(66,82)(73,92)(74,91)(75,96)(76,95)(77,94)(78,93), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,86,10,89)(8,87,11,90)(9,88,12,85)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,52,34,49)(32,53,35,50)(33,54,36,51)(37,47,40,44)(38,48,41,45)(39,43,42,46)(55,73,58,76)(56,74,59,77)(57,75,60,78)(61,72,64,69)(62,67,65,70)(63,68,66,71)(79,94,82,91)(80,95,83,92)(81,96,84,93) );
G=PermutationGroup([[(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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,95,11,93,9,91),(8,96,12,94,10,92),(19,30,23,28,21,26),(20,25,24,29,22,27),(31,40,33,42,35,38),(32,41,34,37,36,39),(43,53,45,49,47,51),(44,54,46,50,48,52),(55,63,57,65,59,61),(56,64,58,66,60,62),(67,74,69,76,71,78),(68,75,70,77,72,73),(79,86,83,90,81,88),(80,87,84,85,82,89)], [(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,69),(8,68),(9,67),(10,72),(11,71),(12,70),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48),(25,50),(26,49),(27,54),(28,53),(29,52),(30,51),(55,88),(56,87),(57,86),(58,85),(59,90),(60,89),(61,81),(62,80),(63,79),(64,84),(65,83),(66,82),(73,92),(74,91),(75,96),(76,95),(77,94),(78,93)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,86,10,89),(8,87,11,90),(9,88,12,85),(13,27,16,30),(14,28,17,25),(15,29,18,26),(31,52,34,49),(32,53,35,50),(33,54,36,51),(37,47,40,44),(38,48,41,45),(39,43,42,46),(55,73,58,76),(56,74,59,77),(57,75,60,78),(61,72,64,69),(62,67,65,70),(63,68,66,71),(79,94,82,91),(80,95,83,92),(81,96,84,93)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | S32 | S3×D4 | D4⋊2S3 | C2×S32 | C4×S32 | D6⋊D6 | D6.4D6 |
| kernel | C62.72C23 | C62.C22 | C3×D6⋊C4 | C4×C3⋊Dic3 | C2×S3×Dic3 | C2×D6⋊S3 | D6⋊S3 | D6⋊C4 | C3⋊Dic3 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | D6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of C62.72C23 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 2 | 11 | 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 | 12 | 12 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,2,0,0,0,0,5,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.72C23 in GAP, Magma, Sage, TeX
C_6^2._{72}C_2^3 % in TeX
G:=Group("C6^2.72C2^3"); // GroupNames label
G:=SmallGroup(288,550);
// by ID
G=gap.SmallGroup(288,550);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,219,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
// generators/relations