metabelian, supersoluble, monomial
Aliases: C62.25C23, (S3×C12)⋊1C4, (C4×S3)⋊3Dic3, C12.101(C4×S3), (C4×Dic3)⋊10S3, (C2×C12).303D6, D6.5(C2×Dic3), C4.24(S3×Dic3), C3⋊5(C42⋊2S3), (Dic3×C12)⋊19C2, C6.22(C4○D12), D6⋊Dic3.15C2, (C2×Dic3).91D6, C12.35(C2×Dic3), (C22×S3).61D6, Dic3⋊Dic3⋊38C2, C6.9(C22×Dic3), (C6×C12).215C22, Dic3.7(C2×Dic3), C32⋊5(C42⋊C2), C2.1(D6.D6), C3⋊1(C23.26D6), (C6×Dic3).141C22, C6.89(S3×C2×C4), (C2×C4).136S32, (S3×C2×C4).10S3, (S3×C2×C12).3C2, C22.23(C2×S32), C2.11(C2×S3×Dic3), (S3×C6).19(C2×C4), (C3×C12).89(C2×C4), (C4×C3⋊Dic3)⋊12C2, (S3×C2×C6).71C22, (C3×C6).13(C4○D4), (C3×C6).51(C22×C4), (C2×C6).44(C22×S3), (C3×Dic3).23(C2×C4), (C2×C3⋊Dic3).117C22, SmallGroup(288,503)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.25C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, de=ed >
Subgroups: 490 in 167 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, S3×C2×C4, C22×C12, S3×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C42⋊2S3, C23.26D6, D6⋊Dic3, Dic3⋊Dic3, Dic3×C12, C4×C3⋊Dic3, S3×C2×C12, C62.25C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C4×S3, C2×Dic3, C22×S3, C42⋊C2, S32, S3×C2×C4, C4○D12, C22×Dic3, S3×Dic3, C2×S32, C42⋊2S3, C23.26D6, D6.D6, C2×S3×Dic3, C62.25C23
►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 33 4 36)(2 32 5 35)(3 31 6 34)(7 69 10 72)(8 68 11 71)(9 67 12 70)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)(25 53 28 50)(26 52 29 49)(27 51 30 54)(55 85 58 88)(56 90 59 87)(57 89 60 86)(61 84 64 81)(62 83 65 80)(63 82 66 79)(73 95 76 92)(74 94 77 91)(75 93 78 96)
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 86 93 81)(8 87 94 82)(9 88 95 83)(10 89 96 84)(11 90 91 79)(12 85 92 80)(31 49 42 43)(32 50 37 44)(33 51 38 45)(34 52 39 46)(35 53 40 47)(36 54 41 48)(55 76 65 67)(56 77 66 68)(57 78 61 69)(58 73 62 70)(59 74 63 71)(60 75 64 72)
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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,69,10,72)(8,68,11,71)(9,67,12,70)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)(25,53,28,50)(26,52,29,49)(27,51,30,54)(55,85,58,88)(56,90,59,87)(57,89,60,86)(61,84,64,81)(62,83,65,80)(63,82,66,79)(73,95,76,92)(74,94,77,91)(75,93,78,96), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,86,93,81)(8,87,94,82)(9,88,95,83)(10,89,96,84)(11,90,91,79)(12,85,92,80)(31,49,42,43)(32,50,37,44)(33,51,38,45)(34,52,39,46)(35,53,40,47)(36,54,41,48)(55,76,65,67)(56,77,66,68)(57,78,61,69)(58,73,62,70)(59,74,63,71)(60,75,64,72)>;
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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,69,10,72)(8,68,11,71)(9,67,12,70)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)(25,53,28,50)(26,52,29,49)(27,51,30,54)(55,85,58,88)(56,90,59,87)(57,89,60,86)(61,84,64,81)(62,83,65,80)(63,82,66,79)(73,95,76,92)(74,94,77,91)(75,93,78,96), (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,86,93,81)(8,87,94,82)(9,88,95,83)(10,89,96,84)(11,90,91,79)(12,85,92,80)(31,49,42,43)(32,50,37,44)(33,51,38,45)(34,52,39,46)(35,53,40,47)(36,54,41,48)(55,76,65,67)(56,77,66,68)(57,78,61,69)(58,73,62,70)(59,74,63,71)(60,75,64,72) );
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)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,69,10,72),(8,68,11,71),(9,67,12,70),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45),(25,53,28,50),(26,52,29,49),(27,51,30,54),(55,85,58,88),(56,90,59,87),(57,89,60,86),(61,84,64,81),(62,83,65,80),(63,82,66,79),(73,95,76,92),(74,94,77,91),(75,93,78,96)], [(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,86,93,81),(8,87,94,82),(9,88,95,83),(10,89,96,84),(11,90,91,79),(12,85,92,80),(31,49,42,43),(32,50,37,44),(33,51,38,45),(34,52,39,46),(35,53,40,47),(36,54,41,48),(55,76,65,67),(56,77,66,68),(57,78,61,69),(58,73,62,70),(59,74,63,71),(60,75,64,72)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 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 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | Dic3 | D6 | D6 | D6 | C4○D4 | C4×S3 | C4○D12 | S32 | S3×Dic3 | C2×S32 | D6.D6 |
| kernel | C62.25C23 | D6⋊Dic3 | Dic3⋊Dic3 | Dic3×C12 | C4×C3⋊Dic3 | S3×C2×C12 | S3×C12 | C4×Dic3 | S3×C2×C4 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C12 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 4 | 3 | 2 | 1 | 4 | 4 | 16 | 1 | 2 | 1 | 4 |
Matrix representation of C62.25C23 ►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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 7 | 0 | 0 | 0 | 0 |
| 7 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 | 12 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,7,0,0,0,0,7,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.25C23 in GAP, Magma, Sage, TeX
C_6^2._{25}C_2^3 % in TeX
G:=Group("C6^2.25C2^3"); // GroupNames label
G:=SmallGroup(288,503);
// by ID
G=gap.SmallGroup(288,503);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=a^3,e^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,d*e=e*d>;
// generators/relations