metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.98D6, C6.532- 1+4, C12⋊Q8⋊13C2, C4⋊C4.312D6, C42⋊3S3⋊6C2, C4.D12⋊13C2, (C4×Dic6)⋊11C2, (C2×C6).77C24, C42⋊C2⋊17S3, C4.98(C4○D12), (C4×C12).28C22, D6⋊C4.84C22, C2.11(Q8○D12), C22⋊C4.101D6, (C22×C4).214D6, C23.8D6⋊4C2, Dic6⋊C4⋊13C2, C12.200(C4○D4), C12.48D4⋊30C2, (C2×C12).698C23, Dic3⋊C4.3C22, C23.98(C22×S3), Dic3.34(C4○D4), (C22×S3).25C23, C4⋊Dic3.293C22, C22.106(S3×C23), (C22×C6).147C23, C23.11D6.1C2, (C2×Dic3).30C23, (C22×C12).234C22, C3⋊1(C22.50C24), (C2×Dic6).232C22, (C4×Dic3).199C22, C6.D4.99C22, C4⋊C4⋊7S3⋊13C2, C6.33(C2×C4○D4), C2.16(S3×C4○D4), (C4×C3⋊D4).6C2, C2.36(C2×C4○D12), (S3×C2×C4).62C22, (C3×C42⋊C2)⋊19C2, (C3×C4⋊C4).313C22, (C2×C4).279(C22×S3), (C2×C3⋊D4).106C22, (C3×C22⋊C4).116C22, SmallGroup(192,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.98D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 472 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, 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, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C22.50C24, C4×Dic6, C42⋊3S3, C23.8D6, C23.11D6, Dic6⋊C4, C12⋊Q8, C4⋊C4⋊7S3, C4.D12, C12.48D4, C4×C3⋊D4, C3×C42⋊C2, C42.98D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.50C24, C2×C4○D12, S3×C4○D4, Q8○D12, C42.98D6
►On 96 points
(1 19 74 25)(2 14 75 32)(3 21 76 27)(4 16 77 34)(5 23 78 29)(6 18 79 36)(7 13 80 31)(8 20 81 26)(9 15 82 33)(10 22 83 28)(11 17 84 35)(12 24 73 30)(37 59 92 61)(38 54 93 68)(39 49 94 63)(40 56 95 70)(41 51 96 65)(42 58 85 72)(43 53 86 67)(44 60 87 62)(45 55 88 69)(46 50 89 64)(47 57 90 71)(48 52 91 66)
(1 92 7 86)(2 93 8 87)(3 94 9 88)(4 95 10 89)(5 96 11 90)(6 85 12 91)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 59 31 53)(26 60 32 54)(27 49 33 55)(28 50 34 56)(29 51 35 57)(30 52 36 58)(37 80 43 74)(38 81 44 75)(39 82 45 76)(40 83 46 77)(41 84 47 78)(42 73 48 79)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 36 31 30)(26 29 32 35)(27 34 33 28)(37 91 43 85)(38 96 44 90)(39 89 45 95)(40 94 46 88)(41 87 47 93)(42 92 48 86)(49 64 55 70)(50 69 56 63)(51 62 57 68)(52 67 58 61)(53 72 59 66)(54 65 60 71)(73 74 79 80)(75 84 81 78)(76 77 82 83)
G:=sub<Sym(96)| (1,19,74,25)(2,14,75,32)(3,21,76,27)(4,16,77,34)(5,23,78,29)(6,18,79,36)(7,13,80,31)(8,20,81,26)(9,15,82,33)(10,22,83,28)(11,17,84,35)(12,24,73,30)(37,59,92,61)(38,54,93,68)(39,49,94,63)(40,56,95,70)(41,51,96,65)(42,58,85,72)(43,53,86,67)(44,60,87,62)(45,55,88,69)(46,50,89,64)(47,57,90,71)(48,52,91,66), (1,92,7,86)(2,93,8,87)(3,94,9,88)(4,95,10,89)(5,96,11,90)(6,85,12,91)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,59,31,53)(26,60,32,54)(27,49,33,55)(28,50,34,56)(29,51,35,57)(30,52,36,58)(37,80,43,74)(38,81,44,75)(39,82,45,76)(40,83,46,77)(41,84,47,78)(42,73,48,79), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,91,43,85)(38,96,44,90)(39,89,45,95)(40,94,46,88)(41,87,47,93)(42,92,48,86)(49,64,55,70)(50,69,56,63)(51,62,57,68)(52,67,58,61)(53,72,59,66)(54,65,60,71)(73,74,79,80)(75,84,81,78)(76,77,82,83)>;
G:=Group( (1,19,74,25)(2,14,75,32)(3,21,76,27)(4,16,77,34)(5,23,78,29)(6,18,79,36)(7,13,80,31)(8,20,81,26)(9,15,82,33)(10,22,83,28)(11,17,84,35)(12,24,73,30)(37,59,92,61)(38,54,93,68)(39,49,94,63)(40,56,95,70)(41,51,96,65)(42,58,85,72)(43,53,86,67)(44,60,87,62)(45,55,88,69)(46,50,89,64)(47,57,90,71)(48,52,91,66), (1,92,7,86)(2,93,8,87)(3,94,9,88)(4,95,10,89)(5,96,11,90)(6,85,12,91)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,59,31,53)(26,60,32,54)(27,49,33,55)(28,50,34,56)(29,51,35,57)(30,52,36,58)(37,80,43,74)(38,81,44,75)(39,82,45,76)(40,83,46,77)(41,84,47,78)(42,73,48,79), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,91,43,85)(38,96,44,90)(39,89,45,95)(40,94,46,88)(41,87,47,93)(42,92,48,86)(49,64,55,70)(50,69,56,63)(51,62,57,68)(52,67,58,61)(53,72,59,66)(54,65,60,71)(73,74,79,80)(75,84,81,78)(76,77,82,83) );
G=PermutationGroup([[(1,19,74,25),(2,14,75,32),(3,21,76,27),(4,16,77,34),(5,23,78,29),(6,18,79,36),(7,13,80,31),(8,20,81,26),(9,15,82,33),(10,22,83,28),(11,17,84,35),(12,24,73,30),(37,59,92,61),(38,54,93,68),(39,49,94,63),(40,56,95,70),(41,51,96,65),(42,58,85,72),(43,53,86,67),(44,60,87,62),(45,55,88,69),(46,50,89,64),(47,57,90,71),(48,52,91,66)], [(1,92,7,86),(2,93,8,87),(3,94,9,88),(4,95,10,89),(5,96,11,90),(6,85,12,91),(13,67,19,61),(14,68,20,62),(15,69,21,63),(16,70,22,64),(17,71,23,65),(18,72,24,66),(25,59,31,53),(26,60,32,54),(27,49,33,55),(28,50,34,56),(29,51,35,57),(30,52,36,58),(37,80,43,74),(38,81,44,75),(39,82,45,76),(40,83,46,77),(41,84,47,78),(42,73,48,79)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,36,31,30),(26,29,32,35),(27,34,33,28),(37,91,43,85),(38,96,44,90),(39,89,45,95),(40,94,46,88),(41,87,47,93),(42,92,48,86),(49,64,55,70),(50,69,56,63),(51,62,57,68),(52,67,58,61),(53,72,59,66),(54,65,60,71),(73,74,79,80),(75,84,81,78),(76,77,82,83)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | ··· | 4S | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.98D6 | C4×Dic6 | C42⋊3S3 | C23.8D6 | C23.11D6 | Dic6⋊C4 | C12⋊Q8 | C4⋊C4⋊7S3 | C4.D12 | C12.48D4 | C4×C3⋊D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.98D6 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 8 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 1 | 8 |
| 1 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,5,0,0,0,2,8],[11,4,0,0,9,2,0,0,0,0,8,0,0,0,0,8],[1,12,0,0,1,0,0,0,0,0,5,1,0,0,0,8],[1,0,0,0,1,12,0,0,0,0,5,0,0,0,0,5] >;
C42.98D6 in GAP, Magma, Sage, TeX
C_4^2._{98}D_6 % in TeX
G:=Group("C4^2.98D6"); // GroupNames label
G:=SmallGroup(192,1092);
// by ID
G=gap.SmallGroup(192,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations