metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.93D6, (C4×D12)⋊9C2, C4⋊C4.270D6, (C4×Dic6)⋊9C2, (S3×C42)⋊18C2, D6⋊Q8⋊50C2, D6.1(C4○D4), Dic3⋊D4.6C2, (C2×C6).72C24, C22⋊C4.96D6, C42⋊C2⋊12S3, C42⋊2S3⋊30C2, D6.D4⋊48C2, C23.9D6⋊52C2, Dic3.Q8⋊44C2, (C22×C4).209D6, C12.255(C4○D4), C4.139(C4○D12), (C2×C12).147C23, (C4×C12).233C22, D6⋊C4.143C22, Dic3.2(C4○D4), C23.8D6⋊48C2, C23.94(C22×S3), C23.11D6⋊48C2, (C2×D12).207C22, C4⋊Dic3.292C22, (C22×C6).142C23, C22.101(S3×C23), (C2×Dic3).25C23, Dic3⋊C4.152C22, (C22×S3).165C23, (C22×C12).377C22, C3⋊2(C23.36C23), (C4×Dic3).196C22, (C2×Dic6).230C22, C6.D4.95C22, (C4×C3⋊D4)⋊52C2, C4⋊C4⋊S3⋊49C2, C6.29(C2×C4○D4), C2.11(S3×C4○D4), C2.31(C2×C4○D12), (S3×C2×C4).290C22, (C3×C42⋊C2)⋊14C2, (C3×C4⋊C4).308C22, (C2×C4).150(C22×S3), (C2×C3⋊D4).102C22, (C3×C22⋊C4).112C22, SmallGroup(192,1087)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.93D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=b-1, dcd-1=c5 >
Subgroups: 552 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C23.36C23, C4×Dic6, S3×C42, C42⋊2S3, C4×D12, C23.8D6, C23.9D6, Dic3⋊D4, C23.11D6, Dic3.Q8, D6.D4, D6⋊Q8, C4⋊C4⋊S3, C4×C3⋊D4, C3×C42⋊C2, C42.93D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C4○D12, S3×C23, C23.36C23, C2×C4○D12, S3×C4○D4, C42.93D6
►On 96 points
(1 17 44 70)(2 18 45 71)(3 19 46 72)(4 20 47 61)(5 21 48 62)(6 22 37 63)(7 23 38 64)(8 24 39 65)(9 13 40 66)(10 14 41 67)(11 15 42 68)(12 16 43 69)(25 55 90 74)(26 56 91 75)(27 57 92 76)(28 58 93 77)(29 59 94 78)(30 60 95 79)(31 49 96 80)(32 50 85 81)(33 51 86 82)(34 52 87 83)(35 53 88 84)(36 54 89 73)
(1 28 38 87)(2 94 39 35)(3 30 40 89)(4 96 41 25)(5 32 42 91)(6 86 43 27)(7 34 44 93)(8 88 45 29)(9 36 46 95)(10 90 47 31)(11 26 48 85)(12 92 37 33)(13 54 72 79)(14 74 61 49)(15 56 62 81)(16 76 63 51)(17 58 64 83)(18 78 65 53)(19 60 66 73)(20 80 67 55)(21 50 68 75)(22 82 69 57)(23 52 70 77)(24 84 71 59)
(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 57 7 51)(2 50 8 56)(3 55 9 49)(4 60 10 54)(5 53 11 59)(6 58 12 52)(13 96 19 90)(14 89 20 95)(15 94 21 88)(16 87 22 93)(17 92 23 86)(18 85 24 91)(25 66 31 72)(26 71 32 65)(27 64 33 70)(28 69 34 63)(29 62 35 68)(30 67 36 61)(37 77 43 83)(38 82 44 76)(39 75 45 81)(40 80 46 74)(41 73 47 79)(42 78 48 84)
G:=sub<Sym(96)| (1,17,44,70)(2,18,45,71)(3,19,46,72)(4,20,47,61)(5,21,48,62)(6,22,37,63)(7,23,38,64)(8,24,39,65)(9,13,40,66)(10,14,41,67)(11,15,42,68)(12,16,43,69)(25,55,90,74)(26,56,91,75)(27,57,92,76)(28,58,93,77)(29,59,94,78)(30,60,95,79)(31,49,96,80)(32,50,85,81)(33,51,86,82)(34,52,87,83)(35,53,88,84)(36,54,89,73), (1,28,38,87)(2,94,39,35)(3,30,40,89)(4,96,41,25)(5,32,42,91)(6,86,43,27)(7,34,44,93)(8,88,45,29)(9,36,46,95)(10,90,47,31)(11,26,48,85)(12,92,37,33)(13,54,72,79)(14,74,61,49)(15,56,62,81)(16,76,63,51)(17,58,64,83)(18,78,65,53)(19,60,66,73)(20,80,67,55)(21,50,68,75)(22,82,69,57)(23,52,70,77)(24,84,71,59), (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), (1,57,7,51)(2,50,8,56)(3,55,9,49)(4,60,10,54)(5,53,11,59)(6,58,12,52)(13,96,19,90)(14,89,20,95)(15,94,21,88)(16,87,22,93)(17,92,23,86)(18,85,24,91)(25,66,31,72)(26,71,32,65)(27,64,33,70)(28,69,34,63)(29,62,35,68)(30,67,36,61)(37,77,43,83)(38,82,44,76)(39,75,45,81)(40,80,46,74)(41,73,47,79)(42,78,48,84)>;
G:=Group( (1,17,44,70)(2,18,45,71)(3,19,46,72)(4,20,47,61)(5,21,48,62)(6,22,37,63)(7,23,38,64)(8,24,39,65)(9,13,40,66)(10,14,41,67)(11,15,42,68)(12,16,43,69)(25,55,90,74)(26,56,91,75)(27,57,92,76)(28,58,93,77)(29,59,94,78)(30,60,95,79)(31,49,96,80)(32,50,85,81)(33,51,86,82)(34,52,87,83)(35,53,88,84)(36,54,89,73), (1,28,38,87)(2,94,39,35)(3,30,40,89)(4,96,41,25)(5,32,42,91)(6,86,43,27)(7,34,44,93)(8,88,45,29)(9,36,46,95)(10,90,47,31)(11,26,48,85)(12,92,37,33)(13,54,72,79)(14,74,61,49)(15,56,62,81)(16,76,63,51)(17,58,64,83)(18,78,65,53)(19,60,66,73)(20,80,67,55)(21,50,68,75)(22,82,69,57)(23,52,70,77)(24,84,71,59), (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), (1,57,7,51)(2,50,8,56)(3,55,9,49)(4,60,10,54)(5,53,11,59)(6,58,12,52)(13,96,19,90)(14,89,20,95)(15,94,21,88)(16,87,22,93)(17,92,23,86)(18,85,24,91)(25,66,31,72)(26,71,32,65)(27,64,33,70)(28,69,34,63)(29,62,35,68)(30,67,36,61)(37,77,43,83)(38,82,44,76)(39,75,45,81)(40,80,46,74)(41,73,47,79)(42,78,48,84) );
G=PermutationGroup([[(1,17,44,70),(2,18,45,71),(3,19,46,72),(4,20,47,61),(5,21,48,62),(6,22,37,63),(7,23,38,64),(8,24,39,65),(9,13,40,66),(10,14,41,67),(11,15,42,68),(12,16,43,69),(25,55,90,74),(26,56,91,75),(27,57,92,76),(28,58,93,77),(29,59,94,78),(30,60,95,79),(31,49,96,80),(32,50,85,81),(33,51,86,82),(34,52,87,83),(35,53,88,84),(36,54,89,73)], [(1,28,38,87),(2,94,39,35),(3,30,40,89),(4,96,41,25),(5,32,42,91),(6,86,43,27),(7,34,44,93),(8,88,45,29),(9,36,46,95),(10,90,47,31),(11,26,48,85),(12,92,37,33),(13,54,72,79),(14,74,61,49),(15,56,62,81),(16,76,63,51),(17,58,64,83),(18,78,65,53),(19,60,66,73),(20,80,67,55),(21,50,68,75),(22,82,69,57),(23,52,70,77),(24,84,71,59)], [(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)], [(1,57,7,51),(2,50,8,56),(3,55,9,49),(4,60,10,54),(5,53,11,59),(6,58,12,52),(13,96,19,90),(14,89,20,95),(15,94,21,88),(16,87,22,93),(17,92,23,86),(18,85,24,91),(25,66,31,72),(26,71,32,65),(27,64,33,70),(28,69,34,63),(29,62,35,68),(30,67,36,61),(37,77,43,83),(38,82,44,76),(39,75,45,81),(40,80,46,74),(41,73,47,79),(42,78,48,84)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | C4○D12 | S3×C4○D4 |
| kernel | C42.93D6 | C4×Dic6 | S3×C42 | C42⋊2S3 | C4×D12 | C23.8D6 | C23.9D6 | Dic3⋊D4 | C23.11D6 | Dic3.Q8 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 4 |
Matrix representation of C42.93D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 11 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 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 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,11,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0] >;
C42.93D6 in GAP, Magma, Sage, TeX
C_4^2._{93}D_6 % in TeX
G:=Group("C4^2.93D6"); // GroupNames label
G:=SmallGroup(192,1087);
// by ID
G=gap.SmallGroup(192,1087);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations