metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.99D6, C6.992+ 1+4, C6.542- 1+4, (C4×D12)⋊11C2, C4⋊C4.274D6, D6⋊Q8⋊6C2, C12⋊2Q8⋊8C2, C42⋊7S3⋊5C2, (C4×Dic6)⋊12C2, C23.9D6⋊5C2, (C2×C6).78C24, D6⋊C4.4C22, C42⋊C2⋊18S3, C2.11(D4○D12), C12⋊7D4.18C2, (C4×C12).29C22, C22⋊C4.102D6, C2.12(Q8○D12), (C22×C4).215D6, C4.120(C4○D12), C12.236(C4○D4), C12.48D4⋊42C2, (C2×C12).151C23, C23.11D6⋊5C2, C4⋊Dic3.36C22, C23.99(C22×S3), (C2×D12).208C22, Dic3⋊C4.75C22, (C22×S3).26C23, C22.107(S3×C23), (C22×C6).148C23, (C2×Dic3).31C23, C6.D4.6C22, (C22×C12).308C22, C3⋊1(C22.36C24), (C2×Dic6).233C22, (C4×Dic3).200C22, C4⋊C4⋊S3⋊6C2, C6.34(C2×C4○D4), C2.37(C2×C4○D12), (S3×C2×C4).195C22, (C3×C42⋊C2)⋊20C2, (C3×C4⋊C4).314C22, (C2×C4).151(C22×S3), (C2×C3⋊D4).11C22, (C3×C22⋊C4).117C22, SmallGroup(192,1093)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.99D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 552 in 216 conjugacy classes, 95 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, 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, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C22.36C24, C4×Dic6, C12⋊2Q8, C4×D12, C42⋊7S3, C23.9D6, C23.11D6, D6⋊Q8, C4⋊C4⋊S3, C12.48D4, C12⋊7D4, C3×C42⋊C2, C42.99D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, C4○D12, S3×C23, C22.36C24, C2×C4○D12, D4○D12, Q8○D12, C42.99D6
►On 96 points
(1 67 19 55)(2 68 20 56)(3 69 21 57)(4 70 22 58)(5 71 23 59)(6 72 24 60)(7 76 28 62)(8 77 29 63)(9 78 30 64)(10 73 25 65)(11 74 26 66)(12 75 27 61)(13 51 81 33)(14 52 82 34)(15 53 83 35)(16 54 84 36)(17 49 79 31)(18 50 80 32)(37 94 46 85)(38 95 47 86)(39 96 48 87)(40 91 43 88)(41 92 44 89)(42 93 45 90)
(1 53 11 43)(2 36 12 41)(3 49 7 45)(4 32 8 37)(5 51 9 47)(6 34 10 39)(13 64 95 59)(14 73 96 72)(15 66 91 55)(16 75 92 68)(17 62 93 57)(18 77 94 70)(19 35 26 40)(20 54 27 44)(21 31 28 42)(22 50 29 46)(23 33 30 38)(24 52 25 48)(56 84 61 89)(58 80 63 85)(60 82 65 87)(67 83 74 88)(69 79 76 90)(71 81 78 86)
(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 11 10)(2 9 12 5)(3 4 7 8)(13 84 95 89)(14 88 96 83)(15 82 91 87)(16 86 92 81)(17 80 93 85)(18 90 94 79)(19 24 26 25)(20 30 27 23)(21 22 28 29)(31 32 42 37)(33 36 38 41)(34 40 39 35)(43 48 53 52)(44 51 54 47)(45 46 49 50)(55 72 66 73)(56 78 61 71)(57 70 62 77)(58 76 63 69)(59 68 64 75)(60 74 65 67)
G:=sub<Sym(96)| (1,67,19,55)(2,68,20,56)(3,69,21,57)(4,70,22,58)(5,71,23,59)(6,72,24,60)(7,76,28,62)(8,77,29,63)(9,78,30,64)(10,73,25,65)(11,74,26,66)(12,75,27,61)(13,51,81,33)(14,52,82,34)(15,53,83,35)(16,54,84,36)(17,49,79,31)(18,50,80,32)(37,94,46,85)(38,95,47,86)(39,96,48,87)(40,91,43,88)(41,92,44,89)(42,93,45,90), (1,53,11,43)(2,36,12,41)(3,49,7,45)(4,32,8,37)(5,51,9,47)(6,34,10,39)(13,64,95,59)(14,73,96,72)(15,66,91,55)(16,75,92,68)(17,62,93,57)(18,77,94,70)(19,35,26,40)(20,54,27,44)(21,31,28,42)(22,50,29,46)(23,33,30,38)(24,52,25,48)(56,84,61,89)(58,80,63,85)(60,82,65,87)(67,83,74,88)(69,79,76,90)(71,81,78,86), (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), (1,6,11,10)(2,9,12,5)(3,4,7,8)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,26,25)(20,30,27,23)(21,22,28,29)(31,32,42,37)(33,36,38,41)(34,40,39,35)(43,48,53,52)(44,51,54,47)(45,46,49,50)(55,72,66,73)(56,78,61,71)(57,70,62,77)(58,76,63,69)(59,68,64,75)(60,74,65,67)>;
G:=Group( (1,67,19,55)(2,68,20,56)(3,69,21,57)(4,70,22,58)(5,71,23,59)(6,72,24,60)(7,76,28,62)(8,77,29,63)(9,78,30,64)(10,73,25,65)(11,74,26,66)(12,75,27,61)(13,51,81,33)(14,52,82,34)(15,53,83,35)(16,54,84,36)(17,49,79,31)(18,50,80,32)(37,94,46,85)(38,95,47,86)(39,96,48,87)(40,91,43,88)(41,92,44,89)(42,93,45,90), (1,53,11,43)(2,36,12,41)(3,49,7,45)(4,32,8,37)(5,51,9,47)(6,34,10,39)(13,64,95,59)(14,73,96,72)(15,66,91,55)(16,75,92,68)(17,62,93,57)(18,77,94,70)(19,35,26,40)(20,54,27,44)(21,31,28,42)(22,50,29,46)(23,33,30,38)(24,52,25,48)(56,84,61,89)(58,80,63,85)(60,82,65,87)(67,83,74,88)(69,79,76,90)(71,81,78,86), (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), (1,6,11,10)(2,9,12,5)(3,4,7,8)(13,84,95,89)(14,88,96,83)(15,82,91,87)(16,86,92,81)(17,80,93,85)(18,90,94,79)(19,24,26,25)(20,30,27,23)(21,22,28,29)(31,32,42,37)(33,36,38,41)(34,40,39,35)(43,48,53,52)(44,51,54,47)(45,46,49,50)(55,72,66,73)(56,78,61,71)(57,70,62,77)(58,76,63,69)(59,68,64,75)(60,74,65,67) );
G=PermutationGroup([[(1,67,19,55),(2,68,20,56),(3,69,21,57),(4,70,22,58),(5,71,23,59),(6,72,24,60),(7,76,28,62),(8,77,29,63),(9,78,30,64),(10,73,25,65),(11,74,26,66),(12,75,27,61),(13,51,81,33),(14,52,82,34),(15,53,83,35),(16,54,84,36),(17,49,79,31),(18,50,80,32),(37,94,46,85),(38,95,47,86),(39,96,48,87),(40,91,43,88),(41,92,44,89),(42,93,45,90)], [(1,53,11,43),(2,36,12,41),(3,49,7,45),(4,32,8,37),(5,51,9,47),(6,34,10,39),(13,64,95,59),(14,73,96,72),(15,66,91,55),(16,75,92,68),(17,62,93,57),(18,77,94,70),(19,35,26,40),(20,54,27,44),(21,31,28,42),(22,50,29,46),(23,33,30,38),(24,52,25,48),(56,84,61,89),(58,80,63,85),(60,82,65,87),(67,83,74,88),(69,79,76,90),(71,81,78,86)], [(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)], [(1,6,11,10),(2,9,12,5),(3,4,7,8),(13,84,95,89),(14,88,96,83),(15,82,91,87),(16,86,92,81),(17,80,93,85),(18,90,94,79),(19,24,26,25),(20,30,27,23),(21,22,28,29),(31,32,42,37),(33,36,38,41),(34,40,39,35),(43,48,53,52),(44,51,54,47),(45,46,49,50),(55,72,66,73),(56,78,61,71),(57,70,62,77),(58,76,63,69),(59,68,64,75),(60,74,65,67)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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○D12 | 2+ 1+4 | 2- 1+4 | D4○D12 | Q8○D12 |
| kernel | C42.99D6 | C4×Dic6 | C12⋊2Q8 | C4×D12 | C42⋊7S3 | C23.9D6 | C23.11D6 | D6⋊Q8 | C4⋊C4⋊S3 | C12.48D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.99D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 | 3 | 6 |
| 0 | 0 | 0 | 4 | 7 | 10 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 5 | 5 | 11 | 12 |
| 0 | 0 | 8 | 8 | 8 | 0 |
| 0 | 0 | 9 | 8 | 8 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 7 | 2 | 1 | 1 |
| 0 | 0 | 12 | 7 | 12 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 7 | 2 | 1 | 1 |
| 0 | 0 | 12 | 7 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,7,9,0,0,0,6,3,0,4,0,0,0,0,3,7,0,0,0,0,6,10],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,8,9,0,0,0,5,8,8,0,0,12,11,8,8,0,0,1,12,0,0],[4,2,0,0,0,0,11,2,0,0,0,0,0,0,0,1,7,12,0,0,12,12,2,7,0,0,0,0,1,12,0,0,0,0,1,0],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,1,0,7,12,0,0,12,12,2,7,0,0,0,0,1,0,0,0,0,0,1,12] >;
C42.99D6 in GAP, Magma, Sage, TeX
C_4^2._{99}D_6 % in TeX
G:=Group("C4^2.99D6"); // GroupNames label
G:=SmallGroup(192,1093);
// by ID
G=gap.SmallGroup(192,1093);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^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^-1>;
// generators/relations