metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.237D6, C4⋊C4.206D6, (S3×C42)⋊11C2, C4.D12⋊36C2, C42.C2⋊16S3, Dic3⋊5D4⋊35C2, C42⋊7S3⋊24C2, D6.10(C4○D4), C12⋊D4.11C2, D6.D4⋊33C2, (C2×C6).235C24, D6⋊C4.60C22, Dic6⋊C4⋊35C2, C12.128(C4○D4), (C4×C12).195C22, (C2×C12).506C23, C4.19(Q8⋊3S3), Dic3.44(C4○D4), (C2×D12).163C22, Dic3⋊C4.51C22, C4⋊Dic3.241C22, C22.256(S3×C23), (C22×S3).101C23, (C4×Dic3).142C22, (C2×Dic3).312C23, (C2×Dic6).179C22, C3⋊10(C23.36C23), C4⋊C4⋊S3⋊33C2, C4⋊C4⋊7S3⋊35C2, C2.86(S3×C4○D4), C6.197(C2×C4○D4), (C3×C42.C2)⋊8C2, (S3×C2×C4).125C22, (C2×C4).79(C22×S3), C2.22(C2×Q8⋊3S3), (C3×C4⋊C4).190C22, SmallGroup(192,1250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.237D6
G = < a,b,c,d | a4=b4=d2=1, c6=a2, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c5 >
Subgroups: 576 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C23.36C23, S3×C42, C42⋊7S3, Dic6⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4, Dic3⋊5D4, D6.D4, C12⋊D4, C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.237D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, Q8⋊3S3, S3×C23, C23.36C23, C2×Q8⋊3S3, S3×C4○D4, C42.237D6
►On 96 points
(1 74 7 80)(2 65 8 71)(3 76 9 82)(4 67 10 61)(5 78 11 84)(6 69 12 63)(13 60 19 54)(14 48 20 42)(15 50 21 56)(16 38 22 44)(17 52 23 58)(18 40 24 46)(25 73 31 79)(26 64 32 70)(27 75 33 81)(28 66 34 72)(29 77 35 83)(30 68 36 62)(37 85 43 91)(39 87 45 93)(41 89 47 95)(49 96 55 90)(51 86 57 92)(53 88 59 94)
(1 53 26 40)(2 60 27 47)(3 55 28 42)(4 50 29 37)(5 57 30 44)(6 52 31 39)(7 59 32 46)(8 54 33 41)(9 49 34 48)(10 56 35 43)(11 51 36 38)(12 58 25 45)(13 81 89 71)(14 76 90 66)(15 83 91 61)(16 78 92 68)(17 73 93 63)(18 80 94 70)(19 75 95 65)(20 82 96 72)(21 77 85 67)(22 84 86 62)(23 79 87 69)(24 74 88 64)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 89)(14 88)(15 87)(16 86)(17 85)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 35)(26 34)(27 33)(28 32)(29 31)(37 45)(38 44)(39 43)(40 42)(46 48)(49 59)(50 58)(51 57)(52 56)(53 55)(61 73)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
G:=sub<Sym(96)| (1,74,7,80)(2,65,8,71)(3,76,9,82)(4,67,10,61)(5,78,11,84)(6,69,12,63)(13,60,19,54)(14,48,20,42)(15,50,21,56)(16,38,22,44)(17,52,23,58)(18,40,24,46)(25,73,31,79)(26,64,32,70)(27,75,33,81)(28,66,34,72)(29,77,35,83)(30,68,36,62)(37,85,43,91)(39,87,45,93)(41,89,47,95)(49,96,55,90)(51,86,57,92)(53,88,59,94), (1,53,26,40)(2,60,27,47)(3,55,28,42)(4,50,29,37)(5,57,30,44)(6,52,31,39)(7,59,32,46)(8,54,33,41)(9,49,34,48)(10,56,35,43)(11,51,36,38)(12,58,25,45)(13,81,89,71)(14,76,90,66)(15,83,91,61)(16,78,92,68)(17,73,93,63)(18,80,94,70)(19,75,95,65)(20,82,96,72)(21,77,85,67)(22,84,86,62)(23,79,87,69)(24,74,88,64), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,89)(14,88)(15,87)(16,86)(17,85)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,35)(26,34)(27,33)(28,32)(29,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,59)(50,58)(51,57)(52,56)(53,55)(61,73)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)>;
G:=Group( (1,74,7,80)(2,65,8,71)(3,76,9,82)(4,67,10,61)(5,78,11,84)(6,69,12,63)(13,60,19,54)(14,48,20,42)(15,50,21,56)(16,38,22,44)(17,52,23,58)(18,40,24,46)(25,73,31,79)(26,64,32,70)(27,75,33,81)(28,66,34,72)(29,77,35,83)(30,68,36,62)(37,85,43,91)(39,87,45,93)(41,89,47,95)(49,96,55,90)(51,86,57,92)(53,88,59,94), (1,53,26,40)(2,60,27,47)(3,55,28,42)(4,50,29,37)(5,57,30,44)(6,52,31,39)(7,59,32,46)(8,54,33,41)(9,49,34,48)(10,56,35,43)(11,51,36,38)(12,58,25,45)(13,81,89,71)(14,76,90,66)(15,83,91,61)(16,78,92,68)(17,73,93,63)(18,80,94,70)(19,75,95,65)(20,82,96,72)(21,77,85,67)(22,84,86,62)(23,79,87,69)(24,74,88,64), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,89)(14,88)(15,87)(16,86)(17,85)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,35)(26,34)(27,33)(28,32)(29,31)(37,45)(38,44)(39,43)(40,42)(46,48)(49,59)(50,58)(51,57)(52,56)(53,55)(61,73)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74) );
G=PermutationGroup([[(1,74,7,80),(2,65,8,71),(3,76,9,82),(4,67,10,61),(5,78,11,84),(6,69,12,63),(13,60,19,54),(14,48,20,42),(15,50,21,56),(16,38,22,44),(17,52,23,58),(18,40,24,46),(25,73,31,79),(26,64,32,70),(27,75,33,81),(28,66,34,72),(29,77,35,83),(30,68,36,62),(37,85,43,91),(39,87,45,93),(41,89,47,95),(49,96,55,90),(51,86,57,92),(53,88,59,94)], [(1,53,26,40),(2,60,27,47),(3,55,28,42),(4,50,29,37),(5,57,30,44),(6,52,31,39),(7,59,32,46),(8,54,33,41),(9,49,34,48),(10,56,35,43),(11,51,36,38),(12,58,25,45),(13,81,89,71),(14,76,90,66),(15,83,91,61),(16,78,92,68),(17,73,93,63),(18,80,94,70),(19,75,95,65),(20,82,96,72),(21,77,85,67),(22,84,86,62),(23,79,87,69),(24,74,88,64)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,89),(14,88),(15,87),(16,86),(17,85),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,90),(25,35),(26,34),(27,33),(28,32),(29,31),(37,45),(38,44),(39,43),(40,42),(46,48),(49,59),(50,58),(51,57),(52,56),(53,55),(61,73),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| 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 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C42.237D6 | S3×C42 | C42⋊7S3 | Dic6⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | C12 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of C42.237D6 ►in GL6(𝔽13)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 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 | 0 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 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 | 12 | 2 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,2,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
C42.237D6 in GAP, Magma, Sage, TeX
C_4^2._{237}D_6 % in TeX
G:=Group("C4^2.237D6"); // GroupNames label
G:=SmallGroup(192,1250);
// by ID
G=gap.SmallGroup(192,1250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,346,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^5>;
// generators/relations