metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.100D6, C6.1002+ 1+4, (C4×D12)⋊12C2, Dic3⋊D4⋊5C2, C4⋊D12⋊4C2, C4⋊C4.275D6, C12⋊7D4⋊42C2, D6.D4⋊5C2, (C2×C6).79C24, C12.6Q8⋊5C2, C42⋊C2⋊19S3, C2.12(D4○D12), (C4×C12).30C22, D6⋊C4.64C22, C22⋊C4.103D6, (C22×C4).216D6, C12.237(C4○D4), C4.121(C4○D12), (C2×C12).152C23, (C2×D12).25C22, Dic3⋊C4.4C22, (C22×S3).27C23, C4⋊Dic3.294C22, C22.108(S3×C23), (C22×C6).149C23, C23.100(C22×S3), (C2×Dic3).32C23, (C22×C12).309C22, C3⋊1(C22.34C24), C6.35(C2×C4○D4), C2.38(C2×C4○D12), (S3×C2×C4).196C22, (C3×C42⋊C2)⋊21C2, (C3×C4⋊C4).315C22, (C2×C4).280(C22×S3), (C2×C3⋊D4).12C22, (C3×C22⋊C4).118C22, SmallGroup(192,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.100D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c5 >
Subgroups: 712 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C22.34C24, C12.6Q8, C4×D12, C4⋊D12, Dic3⋊D4, D6.D4, C12⋊7D4, C3×C42⋊C2, C42.100D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.34C24, C2×C4○D12, D4○D12, C42.100D6
►On 96 points
(1 58 7 52)(2 59 8 53)(3 60 9 54)(4 49 10 55)(5 50 11 56)(6 51 12 57)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 47 31 41)(26 48 32 42)(27 37 33 43)(28 38 34 44)(29 39 35 45)(30 40 36 46)(73 85 79 91)(74 86 80 92)(75 87 81 93)(76 88 82 94)(77 89 83 95)(78 90 84 96)
(1 27 94 18)(2 34 95 13)(3 29 96 20)(4 36 85 15)(5 31 86 22)(6 26 87 17)(7 33 88 24)(8 28 89 19)(9 35 90 14)(10 30 91 21)(11 25 92 16)(12 32 93 23)(37 76 68 58)(38 83 69 53)(39 78 70 60)(40 73 71 55)(41 80 72 50)(42 75 61 57)(43 82 62 52)(44 77 63 59)(45 84 64 54)(46 79 65 49)(47 74 66 56)(48 81 67 51)
(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 88 81)(2 80 89 56)(3 55 90 79)(4 78 91 54)(5 53 92 77)(6 76 93 52)(7 51 94 75)(8 74 95 50)(9 49 96 73)(10 84 85 60)(11 59 86 83)(12 82 87 58)(13 47 28 72)(14 71 29 46)(15 45 30 70)(16 69 31 44)(17 43 32 68)(18 67 33 42)(19 41 34 66)(20 65 35 40)(21 39 36 64)(22 63 25 38)(23 37 26 62)(24 61 27 48)
G:=sub<Sym(96)| (1,58,7,52)(2,59,8,53)(3,60,9,54)(4,49,10,55)(5,50,11,56)(6,51,12,57)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,47,31,41)(26,48,32,42)(27,37,33,43)(28,38,34,44)(29,39,35,45)(30,40,36,46)(73,85,79,91)(74,86,80,92)(75,87,81,93)(76,88,82,94)(77,89,83,95)(78,90,84,96), (1,27,94,18)(2,34,95,13)(3,29,96,20)(4,36,85,15)(5,31,86,22)(6,26,87,17)(7,33,88,24)(8,28,89,19)(9,35,90,14)(10,30,91,21)(11,25,92,16)(12,32,93,23)(37,76,68,58)(38,83,69,53)(39,78,70,60)(40,73,71,55)(41,80,72,50)(42,75,61,57)(43,82,62,52)(44,77,63,59)(45,84,64,54)(46,79,65,49)(47,74,66,56)(48,81,67,51), (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), (1,57,88,81)(2,80,89,56)(3,55,90,79)(4,78,91,54)(5,53,92,77)(6,76,93,52)(7,51,94,75)(8,74,95,50)(9,49,96,73)(10,84,85,60)(11,59,86,83)(12,82,87,58)(13,47,28,72)(14,71,29,46)(15,45,30,70)(16,69,31,44)(17,43,32,68)(18,67,33,42)(19,41,34,66)(20,65,35,40)(21,39,36,64)(22,63,25,38)(23,37,26,62)(24,61,27,48)>;
G:=Group( (1,58,7,52)(2,59,8,53)(3,60,9,54)(4,49,10,55)(5,50,11,56)(6,51,12,57)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,47,31,41)(26,48,32,42)(27,37,33,43)(28,38,34,44)(29,39,35,45)(30,40,36,46)(73,85,79,91)(74,86,80,92)(75,87,81,93)(76,88,82,94)(77,89,83,95)(78,90,84,96), (1,27,94,18)(2,34,95,13)(3,29,96,20)(4,36,85,15)(5,31,86,22)(6,26,87,17)(7,33,88,24)(8,28,89,19)(9,35,90,14)(10,30,91,21)(11,25,92,16)(12,32,93,23)(37,76,68,58)(38,83,69,53)(39,78,70,60)(40,73,71,55)(41,80,72,50)(42,75,61,57)(43,82,62,52)(44,77,63,59)(45,84,64,54)(46,79,65,49)(47,74,66,56)(48,81,67,51), (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), (1,57,88,81)(2,80,89,56)(3,55,90,79)(4,78,91,54)(5,53,92,77)(6,76,93,52)(7,51,94,75)(8,74,95,50)(9,49,96,73)(10,84,85,60)(11,59,86,83)(12,82,87,58)(13,47,28,72)(14,71,29,46)(15,45,30,70)(16,69,31,44)(17,43,32,68)(18,67,33,42)(19,41,34,66)(20,65,35,40)(21,39,36,64)(22,63,25,38)(23,37,26,62)(24,61,27,48) );
G=PermutationGroup([[(1,58,7,52),(2,59,8,53),(3,60,9,54),(4,49,10,55),(5,50,11,56),(6,51,12,57),(13,63,19,69),(14,64,20,70),(15,65,21,71),(16,66,22,72),(17,67,23,61),(18,68,24,62),(25,47,31,41),(26,48,32,42),(27,37,33,43),(28,38,34,44),(29,39,35,45),(30,40,36,46),(73,85,79,91),(74,86,80,92),(75,87,81,93),(76,88,82,94),(77,89,83,95),(78,90,84,96)], [(1,27,94,18),(2,34,95,13),(3,29,96,20),(4,36,85,15),(5,31,86,22),(6,26,87,17),(7,33,88,24),(8,28,89,19),(9,35,90,14),(10,30,91,21),(11,25,92,16),(12,32,93,23),(37,76,68,58),(38,83,69,53),(39,78,70,60),(40,73,71,55),(41,80,72,50),(42,75,61,57),(43,82,62,52),(44,77,63,59),(45,84,64,54),(46,79,65,49),(47,74,66,56),(48,81,67,51)], [(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)], [(1,57,88,81),(2,80,89,56),(3,55,90,79),(4,78,91,54),(5,53,92,77),(6,76,93,52),(7,51,94,75),(8,74,95,50),(9,49,96,73),(10,84,85,60),(11,59,86,83),(12,82,87,58),(13,47,28,72),(14,71,29,46),(15,45,30,70),(16,69,31,44),(17,43,32,68),(18,67,33,42),(19,41,34,66),(20,65,35,40),(21,39,36,64),(22,63,25,38),(23,37,26,62),(24,61,27,48)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | D4○D12 |
| kernel | C42.100D6 | C12.6Q8 | C4×D12 | C4⋊D12 | Dic3⋊D4 | D6.D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 2 | 4 |
Matrix representation of C42.100D6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 7 |
| 0 | 0 | 0 | 0 | 11 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 2 |
| 0 | 0 | 0 | 0 | 6 | 0 | 11 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 6 |
| 0 | 0 | 0 | 0 | 6 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 2 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,11,0,6,0,0,0,0,2,0,6,0,0,0,0,0,0,7,0,11,0,0,0,0,7,0,2,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,2,0,6,0,0,0,0,0,0,11,0,6,0,0,0,0,6,0,11,0,0,0,0,0,0,6,0,2] >;
C42.100D6 in GAP, Magma, Sage, TeX
C_4^2._{100}D_6 % in TeX
G:=Group("C4^2.100D6"); // GroupNames label
G:=SmallGroup(192,1094);
// by ID
G=gap.SmallGroup(192,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations