metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.12D28, C28.47D4, M4(2).2D7, (C2×C4).2D14, (C2×Dic7).C4, C22.5(C4×D7), C4.22(C7⋊D4), C7⋊1(C4.10D4), C4.Dic7.3C2, C2.10(D14⋊C4), C14.9(C22⋊C4), (C2×C28).14C22, (C2×Dic14).6C2, (C7×M4(2)).2C2, (C2×C14).3(C2×C4), SmallGroup(224,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.12D28
G = < a,b,c | a28=1, b4=c2=a14, bab-1=cac-1=a-1, cbc-1=a21b3 >
Smallest permutation representation of C4.12D28
►On 112 points
Generators in S112
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 92 22 99 15 106 8 85)(2 91 23 98 16 105 9 112)(3 90 24 97 17 104 10 111)(4 89 25 96 18 103 11 110)(5 88 26 95 19 102 12 109)(6 87 27 94 20 101 13 108)(7 86 28 93 21 100 14 107)(29 66 36 59 43 80 50 73)(30 65 37 58 44 79 51 72)(31 64 38 57 45 78 52 71)(32 63 39 84 46 77 53 70)(33 62 40 83 47 76 54 69)(34 61 41 82 48 75 55 68)(35 60 42 81 49 74 56 67)
(1 61 15 75)(2 60 16 74)(3 59 17 73)(4 58 18 72)(5 57 19 71)(6 84 20 70)(7 83 21 69)(8 82 22 68)(9 81 23 67)(10 80 24 66)(11 79 25 65)(12 78 26 64)(13 77 27 63)(14 76 28 62)(29 104 43 90)(30 103 44 89)(31 102 45 88)(32 101 46 87)(33 100 47 86)(34 99 48 85)(35 98 49 112)(36 97 50 111)(37 96 51 110)(38 95 52 109)(39 94 53 108)(40 93 54 107)(41 92 55 106)(42 91 56 105)
G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,92,22,99,15,106,8,85)(2,91,23,98,16,105,9,112)(3,90,24,97,17,104,10,111)(4,89,25,96,18,103,11,110)(5,88,26,95,19,102,12,109)(6,87,27,94,20,101,13,108)(7,86,28,93,21,100,14,107)(29,66,36,59,43,80,50,73)(30,65,37,58,44,79,51,72)(31,64,38,57,45,78,52,71)(32,63,39,84,46,77,53,70)(33,62,40,83,47,76,54,69)(34,61,41,82,48,75,55,68)(35,60,42,81,49,74,56,67), (1,61,15,75)(2,60,16,74)(3,59,17,73)(4,58,18,72)(5,57,19,71)(6,84,20,70)(7,83,21,69)(8,82,22,68)(9,81,23,67)(10,80,24,66)(11,79,25,65)(12,78,26,64)(13,77,27,63)(14,76,28,62)(29,104,43,90)(30,103,44,89)(31,102,45,88)(32,101,46,87)(33,100,47,86)(34,99,48,85)(35,98,49,112)(36,97,50,111)(37,96,51,110)(38,95,52,109)(39,94,53,108)(40,93,54,107)(41,92,55,106)(42,91,56,105)>;
G:=Group( (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,92,22,99,15,106,8,85)(2,91,23,98,16,105,9,112)(3,90,24,97,17,104,10,111)(4,89,25,96,18,103,11,110)(5,88,26,95,19,102,12,109)(6,87,27,94,20,101,13,108)(7,86,28,93,21,100,14,107)(29,66,36,59,43,80,50,73)(30,65,37,58,44,79,51,72)(31,64,38,57,45,78,52,71)(32,63,39,84,46,77,53,70)(33,62,40,83,47,76,54,69)(34,61,41,82,48,75,55,68)(35,60,42,81,49,74,56,67), (1,61,15,75)(2,60,16,74)(3,59,17,73)(4,58,18,72)(5,57,19,71)(6,84,20,70)(7,83,21,69)(8,82,22,68)(9,81,23,67)(10,80,24,66)(11,79,25,65)(12,78,26,64)(13,77,27,63)(14,76,28,62)(29,104,43,90)(30,103,44,89)(31,102,45,88)(32,101,46,87)(33,100,47,86)(34,99,48,85)(35,98,49,112)(36,97,50,111)(37,96,51,110)(38,95,52,109)(39,94,53,108)(40,93,54,107)(41,92,55,106)(42,91,56,105) );
G=PermutationGroup([[(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,92,22,99,15,106,8,85),(2,91,23,98,16,105,9,112),(3,90,24,97,17,104,10,111),(4,89,25,96,18,103,11,110),(5,88,26,95,19,102,12,109),(6,87,27,94,20,101,13,108),(7,86,28,93,21,100,14,107),(29,66,36,59,43,80,50,73),(30,65,37,58,44,79,51,72),(31,64,38,57,45,78,52,71),(32,63,39,84,46,77,53,70),(33,62,40,83,47,76,54,69),(34,61,41,82,48,75,55,68),(35,60,42,81,49,74,56,67)], [(1,61,15,75),(2,60,16,74),(3,59,17,73),(4,58,18,72),(5,57,19,71),(6,84,20,70),(7,83,21,69),(8,82,22,68),(9,81,23,67),(10,80,24,66),(11,79,25,65),(12,78,26,64),(13,77,27,63),(14,76,28,62),(29,104,43,90),(30,103,44,89),(31,102,45,88),(32,101,46,87),(33,100,47,86),(34,99,48,85),(35,98,49,112),(36,97,50,111),(37,96,51,110),(38,95,52,109),(39,94,53,108),(40,93,54,107),(41,92,55,106),(42,91,56,105)]])
C4.12D28 is a maximal subgroup of
M4(2).19D14 D28.2D4 D7×C4.10D4 D28.7D4 D4.9D28 D4.10D28 C8.20D28 C8.24D28 M4(2).31D14 D4.3D28 D4.5D28 M4(2).13D14 D28.38D4 M4(2).16D14 D28.40D4
C4.12D28 is a maximal quotient of
C42.2D14 (C2×Dic7)⋊C8 C28.47D8 C28.2D8 M4(2)⋊Dic7
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | D28 | C7⋊D4 | C4×D7 | C4.10D4 | C4.12D28 |
| kernel | C4.12D28 | C4.Dic7 | C7×M4(2) | C2×Dic14 | C2×Dic7 | C28 | M4(2) | C2×C4 | C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C4.12D28 ►in GL4(𝔽113) generated by
| 67 | 81 | 0 | 0 |
| 112 | 100 | 0 | 0 |
| 0 | 0 | 67 | 81 |
| 0 | 0 | 112 | 100 |
| 0 | 0 | 80 | 40 |
| 0 | 0 | 1 | 33 |
| 33 | 81 | 0 | 0 |
| 27 | 80 | 0 | 0 |
| 80 | 40 | 0 | 0 |
| 1 | 33 | 0 | 0 |
| 0 | 0 | 80 | 40 |
| 0 | 0 | 1 | 33 |
G:=sub<GL(4,GF(113))| [67,112,0,0,81,100,0,0,0,0,67,112,0,0,81,100],[0,0,33,27,0,0,81,80,80,1,0,0,40,33,0,0],[80,1,0,0,40,33,0,0,0,0,80,1,0,0,40,33] >;
C4.12D28 in GAP, Magma, Sage, TeX
C_4._{12}D_{28} % in TeX
G:=Group("C4.12D28"); // GroupNames label
G:=SmallGroup(224,30);
// by ID
G=gap.SmallGroup(224,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,121,31,362,86,297,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^3>;
// generators/relations
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