metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.92D6, C6.502- 1+4, C4⋊C4.269D6, C12⋊2Q8⋊7C2, (C2×C4).56D12, C4.72(C2×D12), C42⋊7S3⋊4C2, C4.D12⋊11C2, (C2×C12).202D4, C12.288(C2×D4), (C2×C6).70C24, (C4×C12).8C22, D6⋊C4.2C22, C2.8(Q8○D12), C22⋊C4.94D6, C6.14(C22×D4), C42⋊C2⋊10S3, C2.16(C22×D12), C22.21(C2×D12), (C22×C4).207D6, (C2×C12).145C23, (C22×Dic6)⋊15C2, C23.21D6⋊4C2, C4⋊Dic3.33C22, C22.99(S3×C23), (C2×D12).206C22, (C22×S3).20C23, C23.168(C22×S3), (C22×C6).140C23, (C2×Dic3).24C23, (C22×C12).230C22, C3⋊1(C23.38C23), (C2×Dic6).285C22, (C22×Dic3).87C22, (C2×C6).51(C2×D4), (S3×C2×C4).59C22, (C2×C4○D12).19C2, (C3×C42⋊C2)⋊12C2, (C3×C4⋊C4).307C22, (C2×C4).576(C22×S3), (C2×C3⋊D4).101C22, (C3×C22⋊C4).102C22, SmallGroup(192,1085)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.92D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=b-1, dcd-1=c5 >
Subgroups: 680 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C23.38C23, C12⋊2Q8, C42⋊7S3, C23.21D6, C4.D12, C3×C42⋊C2, C22×Dic6, C2×C4○D12, C42.92D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, 2- 1+4, C2×D12, S3×C23, C23.38C23, C22×D12, Q8○D12, C42.92D6
►On 96 points
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 86 19 92)(14 87 20 93)(15 88 21 94)(16 89 22 95)(17 90 23 96)(18 91 24 85)(37 84 43 78)(38 73 44 79)(39 74 45 80)(40 75 46 81)(41 76 47 82)(42 77 48 83)(49 64 55 70)(50 65 56 71)(51 66 57 72)(52 67 58 61)(53 68 59 62)(54 69 60 63)
(1 48 49 95)(2 43 50 90)(3 38 51 85)(4 45 52 92)(5 40 53 87)(6 47 54 94)(7 42 55 89)(8 37 56 96)(9 44 57 91)(10 39 58 86)(11 46 59 93)(12 41 60 88)(13 34 80 67)(14 29 81 62)(15 36 82 69)(16 31 83 64)(17 26 84 71)(18 33 73 66)(19 28 74 61)(20 35 75 68)(21 30 76 63)(22 25 77 70)(23 32 78 65)(24 27 79 72)
(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 30 7 36)(2 35 8 29)(3 28 9 34)(4 33 10 27)(5 26 11 32)(6 31 12 25)(13 38 19 44)(14 43 20 37)(15 48 21 42)(16 41 22 47)(17 46 23 40)(18 39 24 45)(49 63 55 69)(50 68 56 62)(51 61 57 67)(52 66 58 72)(53 71 59 65)(54 64 60 70)(73 86 79 92)(74 91 80 85)(75 96 81 90)(76 89 82 95)(77 94 83 88)(78 87 84 93)
G:=sub<Sym(96)| (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,86,19,92)(14,87,20,93)(15,88,21,94)(16,89,22,95)(17,90,23,96)(18,91,24,85)(37,84,43,78)(38,73,44,79)(39,74,45,80)(40,75,46,81)(41,76,47,82)(42,77,48,83)(49,64,55,70)(50,65,56,71)(51,66,57,72)(52,67,58,61)(53,68,59,62)(54,69,60,63), (1,48,49,95)(2,43,50,90)(3,38,51,85)(4,45,52,92)(5,40,53,87)(6,47,54,94)(7,42,55,89)(8,37,56,96)(9,44,57,91)(10,39,58,86)(11,46,59,93)(12,41,60,88)(13,34,80,67)(14,29,81,62)(15,36,82,69)(16,31,83,64)(17,26,84,71)(18,33,73,66)(19,28,74,61)(20,35,75,68)(21,30,76,63)(22,25,77,70)(23,32,78,65)(24,27,79,72), (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), (1,30,7,36)(2,35,8,29)(3,28,9,34)(4,33,10,27)(5,26,11,32)(6,31,12,25)(13,38,19,44)(14,43,20,37)(15,48,21,42)(16,41,22,47)(17,46,23,40)(18,39,24,45)(49,63,55,69)(50,68,56,62)(51,61,57,67)(52,66,58,72)(53,71,59,65)(54,64,60,70)(73,86,79,92)(74,91,80,85)(75,96,81,90)(76,89,82,95)(77,94,83,88)(78,87,84,93)>;
G:=Group( (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,86,19,92)(14,87,20,93)(15,88,21,94)(16,89,22,95)(17,90,23,96)(18,91,24,85)(37,84,43,78)(38,73,44,79)(39,74,45,80)(40,75,46,81)(41,76,47,82)(42,77,48,83)(49,64,55,70)(50,65,56,71)(51,66,57,72)(52,67,58,61)(53,68,59,62)(54,69,60,63), (1,48,49,95)(2,43,50,90)(3,38,51,85)(4,45,52,92)(5,40,53,87)(6,47,54,94)(7,42,55,89)(8,37,56,96)(9,44,57,91)(10,39,58,86)(11,46,59,93)(12,41,60,88)(13,34,80,67)(14,29,81,62)(15,36,82,69)(16,31,83,64)(17,26,84,71)(18,33,73,66)(19,28,74,61)(20,35,75,68)(21,30,76,63)(22,25,77,70)(23,32,78,65)(24,27,79,72), (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), (1,30,7,36)(2,35,8,29)(3,28,9,34)(4,33,10,27)(5,26,11,32)(6,31,12,25)(13,38,19,44)(14,43,20,37)(15,48,21,42)(16,41,22,47)(17,46,23,40)(18,39,24,45)(49,63,55,69)(50,68,56,62)(51,61,57,67)(52,66,58,72)(53,71,59,65)(54,64,60,70)(73,86,79,92)(74,91,80,85)(75,96,81,90)(76,89,82,95)(77,94,83,88)(78,87,84,93) );
G=PermutationGroup([[(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,86,19,92),(14,87,20,93),(15,88,21,94),(16,89,22,95),(17,90,23,96),(18,91,24,85),(37,84,43,78),(38,73,44,79),(39,74,45,80),(40,75,46,81),(41,76,47,82),(42,77,48,83),(49,64,55,70),(50,65,56,71),(51,66,57,72),(52,67,58,61),(53,68,59,62),(54,69,60,63)], [(1,48,49,95),(2,43,50,90),(3,38,51,85),(4,45,52,92),(5,40,53,87),(6,47,54,94),(7,42,55,89),(8,37,56,96),(9,44,57,91),(10,39,58,86),(11,46,59,93),(12,41,60,88),(13,34,80,67),(14,29,81,62),(15,36,82,69),(16,31,83,64),(17,26,84,71),(18,33,73,66),(19,28,74,61),(20,35,75,68),(21,30,76,63),(22,25,77,70),(23,32,78,65),(24,27,79,72)], [(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)], [(1,30,7,36),(2,35,8,29),(3,28,9,34),(4,33,10,27),(5,26,11,32),(6,31,12,25),(13,38,19,44),(14,43,20,37),(15,48,21,42),(16,41,22,47),(17,46,23,40),(18,39,24,45),(49,63,55,69),(50,68,56,62),(51,61,57,67),(52,66,58,72),(53,71,59,65),(54,64,60,70),(73,86,79,92),(74,91,80,85),(75,96,81,90),(76,89,82,95),(77,94,83,88),(78,87,84,93)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D12 | 2- 1+4 | Q8○D12 |
| kernel | C42.92D6 | C12⋊2Q8 | C42⋊7S3 | C23.21D6 | C4.D12 | C3×C42⋊C2 | C22×Dic6 | C2×C4○D12 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
Matrix representation of C42.92D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 2 | 0 |
| 0 | 0 | 9 | 11 | 0 | 2 |
| 0 | 0 | 12 | 0 | 11 | 9 |
| 0 | 0 | 0 | 12 | 4 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 8 | 8 |
| 0 | 0 | 2 | 10 | 5 | 0 |
| 0 | 0 | 6 | 7 | 5 | 2 |
| 0 | 0 | 6 | 12 | 11 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 11 | 2 |
| 0 | 0 | 8 | 10 | 4 | 2 |
| 0 | 0 | 12 | 1 | 11 | 3 |
| 0 | 0 | 2 | 1 | 5 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[1,3,0,0,0,0,8,12,0,0,0,0,0,0,2,9,12,0,0,0,4,11,0,12,0,0,2,0,11,4,0,0,0,2,9,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,2,6,6,0,0,11,10,7,12,0,0,8,5,5,11,0,0,8,0,2,3],[12,10,0,0,0,0,0,1,0,0,0,0,0,0,3,8,12,2,0,0,11,10,1,1,0,0,11,4,11,5,0,0,2,2,3,2] >;
C42.92D6 in GAP, Magma, Sage, TeX
C_4^2._{92}D_6 % in TeX
G:=Group("C4^2.92D6"); // GroupNames label
G:=SmallGroup(192,1085);
// by ID
G=gap.SmallGroup(192,1085);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations