metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.41D4, C12.58D8, D8.1Dic3, Q16.1Dic3, C4○D8.1S3, (C3×D8).1C4, C24.15(C2×C4), (C2×C8).250D6, (C3×Q16).1C4, C24.C4⋊5C2, C8.9(C2×Dic3), C4.31(D4⋊S3), (C2×C12).119D4, C3⋊3(D8.C4), C8.31(C3⋊D4), (C2×C6).12SD16, (C2×C24).38C22, C6.31(D4⋊C4), C12.18(C22⋊C4), C4.6(C6.D4), C22.1(D4.S3), C2.11(D4⋊Dic3), (C2×C3⋊C16)⋊2C2, (C3×C4○D8).1C2, (C2×C4).121(C3⋊D4), SmallGroup(192,126)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.41D4
G = < a,b,c | a12=1, b8=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b7 >
Smallest permutation representation of C24.41D4
►On 96 points
Generators in S96
(1 44 32 94 60 71 9 36 24 86 52 79)(2 72 53 95 25 45 10 80 61 87 17 37)(3 46 18 96 62 73 11 38 26 88 54 65)(4 74 55 81 27 47 12 66 63 89 19 39)(5 48 20 82 64 75 13 40 28 90 56 67)(6 76 57 83 29 33 14 68 49 91 21 41)(7 34 22 84 50 77 15 42 30 92 58 69)(8 78 59 85 31 35 16 70 51 93 23 43)
(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 85 86 8 9 93 94 16)(2 7 87 92 10 15 95 84)(3 91 88 14 11 83 96 6)(4 13 89 82 12 5 81 90)(17 22 80 69 25 30 72 77)(18 68 65 29 26 76 73 21)(19 28 66 75 27 20 74 67)(23 24 70 71 31 32 78 79)(33 46 57 54 41 38 49 62)(34 53 58 37 42 61 50 45)(35 36 59 60 43 44 51 52)(39 48 63 56 47 40 55 64)
G:=sub<Sym(96)| (1,44,32,94,60,71,9,36,24,86,52,79)(2,72,53,95,25,45,10,80,61,87,17,37)(3,46,18,96,62,73,11,38,26,88,54,65)(4,74,55,81,27,47,12,66,63,89,19,39)(5,48,20,82,64,75,13,40,28,90,56,67)(6,76,57,83,29,33,14,68,49,91,21,41)(7,34,22,84,50,77,15,42,30,92,58,69)(8,78,59,85,31,35,16,70,51,93,23,43), (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), (1,85,86,8,9,93,94,16)(2,7,87,92,10,15,95,84)(3,91,88,14,11,83,96,6)(4,13,89,82,12,5,81,90)(17,22,80,69,25,30,72,77)(18,68,65,29,26,76,73,21)(19,28,66,75,27,20,74,67)(23,24,70,71,31,32,78,79)(33,46,57,54,41,38,49,62)(34,53,58,37,42,61,50,45)(35,36,59,60,43,44,51,52)(39,48,63,56,47,40,55,64)>;
G:=Group( (1,44,32,94,60,71,9,36,24,86,52,79)(2,72,53,95,25,45,10,80,61,87,17,37)(3,46,18,96,62,73,11,38,26,88,54,65)(4,74,55,81,27,47,12,66,63,89,19,39)(5,48,20,82,64,75,13,40,28,90,56,67)(6,76,57,83,29,33,14,68,49,91,21,41)(7,34,22,84,50,77,15,42,30,92,58,69)(8,78,59,85,31,35,16,70,51,93,23,43), (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), (1,85,86,8,9,93,94,16)(2,7,87,92,10,15,95,84)(3,91,88,14,11,83,96,6)(4,13,89,82,12,5,81,90)(17,22,80,69,25,30,72,77)(18,68,65,29,26,76,73,21)(19,28,66,75,27,20,74,67)(23,24,70,71,31,32,78,79)(33,46,57,54,41,38,49,62)(34,53,58,37,42,61,50,45)(35,36,59,60,43,44,51,52)(39,48,63,56,47,40,55,64) );
G=PermutationGroup([[(1,44,32,94,60,71,9,36,24,86,52,79),(2,72,53,95,25,45,10,80,61,87,17,37),(3,46,18,96,62,73,11,38,26,88,54,65),(4,74,55,81,27,47,12,66,63,89,19,39),(5,48,20,82,64,75,13,40,28,90,56,67),(6,76,57,83,29,33,14,68,49,91,21,41),(7,34,22,84,50,77,15,42,30,92,58,69),(8,78,59,85,31,35,16,70,51,93,23,43)], [(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)], [(1,85,86,8,9,93,94,16),(2,7,87,92,10,15,95,84),(3,91,88,14,11,83,96,6),(4,13,89,82,12,5,81,90),(17,22,80,69,25,30,72,77),(18,68,65,29,26,76,73,21),(19,28,66,75,27,20,74,67),(23,24,70,71,31,32,78,79),(33,46,57,54,41,38,49,62),(34,53,58,37,42,61,50,45),(35,36,59,60,43,44,51,52),(39,48,63,56,47,40,55,64)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 2 | 8 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 4 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | Dic3 | Dic3 | D8 | SD16 | C3⋊D4 | C3⋊D4 | D8.C4 | D4⋊S3 | D4.S3 | C24.41D4 |
| kernel | C24.41D4 | C2×C3⋊C16 | C24.C4 | C3×C4○D8 | C3×D8 | C3×Q16 | C4○D8 | C24 | C2×C12 | C2×C8 | D8 | Q16 | C12 | C2×C6 | C8 | C2×C4 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C24.41D4 ►in GL4(𝔽97) generated by
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 35 | 0 |
| 0 | 0 | 46 | 61 |
| 46 | 67 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 36 | 86 |
| 46 | 67 | 0 | 0 |
| 31 | 51 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 37 | 86 |
G:=sub<GL(4,GF(97))| [22,0,0,0,0,22,0,0,0,0,35,46,0,0,0,61],[46,15,0,0,67,16,0,0,0,0,11,36,0,0,2,86],[46,31,0,0,67,51,0,0,0,0,11,37,0,0,2,86] >;
C24.41D4 in GAP, Magma, Sage, TeX
C_{24}._{41}D_4 % in TeX
G:=Group("C24.41D4"); // GroupNames label
G:=SmallGroup(192,126);
// by ID
G=gap.SmallGroup(192,126);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,184,675,346,192,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^8=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^7>;
// generators/relations
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