metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D28, C42⋊6D7, C28.28D4, (C4×C28)⋊5C2, D14⋊C4⋊1C2, C2.6(C2×D28), C14.4(C2×D4), (C2×D28).2C2, (C2×C4).77D14, C7⋊1(C4.4D4), (C2×Dic14)⋊1C2, C2.7(C4○D28), C14.5(C4○D4), (C2×C14).16C23, (C2×C28).74C22, (C2×Dic7).3C22, (C22×D7).2C22, C22.37(C22×D7), SmallGroup(224,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D28
G = < a,b,c | a4=b28=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 374 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4.4D4, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, D14⋊C4, C4×C28, C2×Dic14, C2×D28, C4.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, D28, C22×D7, C2×D28, C4○D28, C4.D28
►On 112 points
(1 34 102 81)(2 35 103 82)(3 36 104 83)(4 37 105 84)(5 38 106 57)(6 39 107 58)(7 40 108 59)(8 41 109 60)(9 42 110 61)(10 43 111 62)(11 44 112 63)(12 45 85 64)(13 46 86 65)(14 47 87 66)(15 48 88 67)(16 49 89 68)(17 50 90 69)(18 51 91 70)(19 52 92 71)(20 53 93 72)(21 54 94 73)(22 55 95 74)(23 56 96 75)(24 29 97 76)(25 30 98 77)(26 31 99 78)(27 32 100 79)(28 33 101 80)
(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)
(1 101 102 28)(2 27 103 100)(3 99 104 26)(4 25 105 98)(5 97 106 24)(6 23 107 96)(7 95 108 22)(8 21 109 94)(9 93 110 20)(10 19 111 92)(11 91 112 18)(12 17 85 90)(13 89 86 16)(14 15 87 88)(29 57 76 38)(30 37 77 84)(31 83 78 36)(32 35 79 82)(33 81 80 34)(39 75 58 56)(40 55 59 74)(41 73 60 54)(42 53 61 72)(43 71 62 52)(44 51 63 70)(45 69 64 50)(46 49 65 68)(47 67 66 48)
G:=sub<Sym(112)| (1,34,102,81)(2,35,103,82)(3,36,104,83)(4,37,105,84)(5,38,106,57)(6,39,107,58)(7,40,108,59)(8,41,109,60)(9,42,110,61)(10,43,111,62)(11,44,112,63)(12,45,85,64)(13,46,86,65)(14,47,87,66)(15,48,88,67)(16,49,89,68)(17,50,90,69)(18,51,91,70)(19,52,92,71)(20,53,93,72)(21,54,94,73)(22,55,95,74)(23,56,96,75)(24,29,97,76)(25,30,98,77)(26,31,99,78)(27,32,100,79)(28,33,101,80), (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), (1,101,102,28)(2,27,103,100)(3,99,104,26)(4,25,105,98)(5,97,106,24)(6,23,107,96)(7,95,108,22)(8,21,109,94)(9,93,110,20)(10,19,111,92)(11,91,112,18)(12,17,85,90)(13,89,86,16)(14,15,87,88)(29,57,76,38)(30,37,77,84)(31,83,78,36)(32,35,79,82)(33,81,80,34)(39,75,58,56)(40,55,59,74)(41,73,60,54)(42,53,61,72)(43,71,62,52)(44,51,63,70)(45,69,64,50)(46,49,65,68)(47,67,66,48)>;
G:=Group( (1,34,102,81)(2,35,103,82)(3,36,104,83)(4,37,105,84)(5,38,106,57)(6,39,107,58)(7,40,108,59)(8,41,109,60)(9,42,110,61)(10,43,111,62)(11,44,112,63)(12,45,85,64)(13,46,86,65)(14,47,87,66)(15,48,88,67)(16,49,89,68)(17,50,90,69)(18,51,91,70)(19,52,92,71)(20,53,93,72)(21,54,94,73)(22,55,95,74)(23,56,96,75)(24,29,97,76)(25,30,98,77)(26,31,99,78)(27,32,100,79)(28,33,101,80), (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), (1,101,102,28)(2,27,103,100)(3,99,104,26)(4,25,105,98)(5,97,106,24)(6,23,107,96)(7,95,108,22)(8,21,109,94)(9,93,110,20)(10,19,111,92)(11,91,112,18)(12,17,85,90)(13,89,86,16)(14,15,87,88)(29,57,76,38)(30,37,77,84)(31,83,78,36)(32,35,79,82)(33,81,80,34)(39,75,58,56)(40,55,59,74)(41,73,60,54)(42,53,61,72)(43,71,62,52)(44,51,63,70)(45,69,64,50)(46,49,65,68)(47,67,66,48) );
G=PermutationGroup([[(1,34,102,81),(2,35,103,82),(3,36,104,83),(4,37,105,84),(5,38,106,57),(6,39,107,58),(7,40,108,59),(8,41,109,60),(9,42,110,61),(10,43,111,62),(11,44,112,63),(12,45,85,64),(13,46,86,65),(14,47,87,66),(15,48,88,67),(16,49,89,68),(17,50,90,69),(18,51,91,70),(19,52,92,71),(20,53,93,72),(21,54,94,73),(22,55,95,74),(23,56,96,75),(24,29,97,76),(25,30,98,77),(26,31,99,78),(27,32,100,79),(28,33,101,80)], [(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)], [(1,101,102,28),(2,27,103,100),(3,99,104,26),(4,25,105,98),(5,97,106,24),(6,23,107,96),(7,95,108,22),(8,21,109,94),(9,93,110,20),(10,19,111,92),(11,91,112,18),(12,17,85,90),(13,89,86,16),(14,15,87,88),(29,57,76,38),(30,37,77,84),(31,83,78,36),(32,35,79,82),(33,81,80,34),(39,75,58,56),(40,55,59,74),(41,73,60,54),(42,53,61,72),(43,71,62,52),(44,51,63,70),(45,69,64,50),(46,49,65,68),(47,67,66,48)]])
C4.D28 is a maximal subgroup of
C42.D14 C8.8D28 C42.264D14 C8⋊D28 C42.20D14 C8.D28 M4(2)⋊D14 D4.10D28 D28.19D4 C42.36D14 D4.1D28 Q8.1D28 C42.214D14 C42.216D14 C42.74D14 C42.80D14 C42.82D14 C42.276D14 C42.277D14 C42⋊9D14 C42.92D14 C42⋊10D14 C42.97D14 C42.99D14 D28⋊23D4 Dic14⋊23D4 D4⋊5D28 C42.114D14 C42⋊17D14 C42.122D14 Q8⋊5D28 C42.133D14 C42.135D14 C42.136D14 C42.233D14 D7×C4.4D4 C42⋊22D14 C42.145D14 C42.237D14 C42.157D14 C42.158D14 C42⋊23D14 C42.164D14 C42⋊26D14 C42.171D14 C42.178D14
C4.D28 is a maximal quotient of
(C2×C28).28D4 (C2×Dic7)⋊3D4 (C2×C4).20D28 (C2×C4).21D28 C28.14Q16 C4.5D56 C42.264D14 C42.14D14 C42.19D14 C42.20D14 (C2×C28)⋊10Q8 C42⋊9Dic7 (C2×C4)⋊6D28 (C2×C42)⋊D7
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | ··· | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D28 | C4○D28 |
| kernel | C4.D28 | D14⋊C4 | C4×C28 | C2×Dic14 | C2×D28 | C28 | C42 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 4 | 9 | 12 | 24 |
Matrix representation of C4.D28 ►in GL4(𝔽29) generated by
| 7 | 5 | 0 | 0 |
| 19 | 22 | 0 | 0 |
| 0 | 0 | 18 | 11 |
| 0 | 0 | 10 | 11 |
| 28 | 26 | 0 | 0 |
| 6 | 19 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 23 | 22 | 0 | 0 |
| 26 | 6 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 24 | 17 |
G:=sub<GL(4,GF(29))| [7,19,0,0,5,22,0,0,0,0,18,10,0,0,11,11],[28,6,0,0,26,19,0,0,0,0,12,0,0,0,0,12],[23,26,0,0,22,6,0,0,0,0,12,24,0,0,0,17] >;
C4.D28 in GAP, Magma, Sage, TeX
C_4.D_{28} % in TeX
G:=Group("C4.D28"); // GroupNames label
G:=SmallGroup(224,70);
// by ID
G=gap.SmallGroup(224,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,218,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^4=b^28=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations