metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.3C42, C23.33D28, M4(2)⋊3Dic7, C14.12C4≀C2, C4⋊Dic7⋊11C4, (C4×Dic7)⋊6C4, C28.10(C4⋊C4), (C2×C28).10Q8, C4.3(C4×Dic7), C7⋊3(C42⋊6C4), (C2×C28).492D4, (C7×M4(2))⋊6C4, (C2×C4).25Dic14, (C22×C14).46D4, (C2×M4(2)).6D7, C2.3(D28⋊4C4), C28.94(C22⋊C4), C4.10(Dic7⋊C4), (C22×C4).325D14, C22.3(C4⋊Dic7), C4.27(C23.D7), C22.42(D14⋊C4), (C14×M4(2)).10C2, (C22×C28).124C22, C23.21D14.8C2, C14.14(C2.C42), C2.14(C14.C42), (C2×C4).69(C4×D7), (C2×C4×Dic7).2C2, (C2×C14).7(C4⋊C4), (C2×C28).62(C2×C4), (C2×C4).39(C2×Dic7), (C2×C4).179(C7⋊D4), (C2×C14).54(C22⋊C4), SmallGroup(448,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.3C42
G = < a,b,c | a28=b4=1, c4=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b >
Subgroups: 452 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C42, C42⋊C2, C2×M4(2), C56, C2×Dic7, C2×C28, C22×C14, C42⋊6C4, C4×Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C7×M4(2), C22×Dic7, C22×C28, C2×C4×Dic7, C23.21D14, C14×M4(2), C28.3C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4≀C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C42⋊6C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, D28⋊4C4, C14.C42, C28.3C42
►On 112 points
(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 81 29 94)(2 80 30 93)(3 79 31 92)(4 78 32 91)(5 77 33 90)(6 76 34 89)(7 75 35 88)(8 74 36 87)(9 73 37 86)(10 72 38 85)(11 71 39 112)(12 70 40 111)(13 69 41 110)(14 68 42 109)(15 67 43 108)(16 66 44 107)(17 65 45 106)(18 64 46 105)(19 63 47 104)(20 62 48 103)(21 61 49 102)(22 60 50 101)(23 59 51 100)(24 58 52 99)(25 57 53 98)(26 84 54 97)(27 83 55 96)(28 82 56 95)
(1 94 36 74 15 108 50 60)(2 109 37 61 16 95 51 75)(3 96 38 76 17 110 52 62)(4 111 39 63 18 97 53 77)(5 98 40 78 19 112 54 64)(6 85 41 65 20 99 55 79)(7 100 42 80 21 86 56 66)(8 87 43 67 22 101 29 81)(9 102 44 82 23 88 30 68)(10 89 45 69 24 103 31 83)(11 104 46 84 25 90 32 70)(12 91 47 71 26 105 33 57)(13 106 48 58 27 92 34 72)(14 93 49 73 28 107 35 59)
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), (1,81,29,94)(2,80,30,93)(3,79,31,92)(4,78,32,91)(5,77,33,90)(6,76,34,89)(7,75,35,88)(8,74,36,87)(9,73,37,86)(10,72,38,85)(11,71,39,112)(12,70,40,111)(13,69,41,110)(14,68,42,109)(15,67,43,108)(16,66,44,107)(17,65,45,106)(18,64,46,105)(19,63,47,104)(20,62,48,103)(21,61,49,102)(22,60,50,101)(23,59,51,100)(24,58,52,99)(25,57,53,98)(26,84,54,97)(27,83,55,96)(28,82,56,95), (1,94,36,74,15,108,50,60)(2,109,37,61,16,95,51,75)(3,96,38,76,17,110,52,62)(4,111,39,63,18,97,53,77)(5,98,40,78,19,112,54,64)(6,85,41,65,20,99,55,79)(7,100,42,80,21,86,56,66)(8,87,43,67,22,101,29,81)(9,102,44,82,23,88,30,68)(10,89,45,69,24,103,31,83)(11,104,46,84,25,90,32,70)(12,91,47,71,26,105,33,57)(13,106,48,58,27,92,34,72)(14,93,49,73,28,107,35,59)>;
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), (1,81,29,94)(2,80,30,93)(3,79,31,92)(4,78,32,91)(5,77,33,90)(6,76,34,89)(7,75,35,88)(8,74,36,87)(9,73,37,86)(10,72,38,85)(11,71,39,112)(12,70,40,111)(13,69,41,110)(14,68,42,109)(15,67,43,108)(16,66,44,107)(17,65,45,106)(18,64,46,105)(19,63,47,104)(20,62,48,103)(21,61,49,102)(22,60,50,101)(23,59,51,100)(24,58,52,99)(25,57,53,98)(26,84,54,97)(27,83,55,96)(28,82,56,95), (1,94,36,74,15,108,50,60)(2,109,37,61,16,95,51,75)(3,96,38,76,17,110,52,62)(4,111,39,63,18,97,53,77)(5,98,40,78,19,112,54,64)(6,85,41,65,20,99,55,79)(7,100,42,80,21,86,56,66)(8,87,43,67,22,101,29,81)(9,102,44,82,23,88,30,68)(10,89,45,69,24,103,31,83)(11,104,46,84,25,90,32,70)(12,91,47,71,26,105,33,57)(13,106,48,58,27,92,34,72)(14,93,49,73,28,107,35,59) );
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)], [(1,81,29,94),(2,80,30,93),(3,79,31,92),(4,78,32,91),(5,77,33,90),(6,76,34,89),(7,75,35,88),(8,74,36,87),(9,73,37,86),(10,72,38,85),(11,71,39,112),(12,70,40,111),(13,69,41,110),(14,68,42,109),(15,67,43,108),(16,66,44,107),(17,65,45,106),(18,64,46,105),(19,63,47,104),(20,62,48,103),(21,61,49,102),(22,60,50,101),(23,59,51,100),(24,58,52,99),(25,57,53,98),(26,84,54,97),(27,83,55,96),(28,82,56,95)], [(1,94,36,74,15,108,50,60),(2,109,37,61,16,95,51,75),(3,96,38,76,17,110,52,62),(4,111,39,63,18,97,53,77),(5,98,40,78,19,112,54,64),(6,85,41,65,20,99,55,79),(7,100,42,80,21,86,56,66),(8,87,43,67,22,101,29,81),(9,102,44,82,23,88,30,68),(10,89,45,69,24,103,31,83),(11,104,46,84,25,90,32,70),(12,91,47,71,26,105,33,57),(13,106,48,58,27,92,34,72),(14,93,49,73,28,107,35,59)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D7 | Dic7 | D14 | C4≀C2 | Dic14 | C4×D7 | C7⋊D4 | D28 | D28⋊4C4 |
| kernel | C28.3C42 | C2×C4×Dic7 | C23.21D14 | C14×M4(2) | C4×Dic7 | C4⋊Dic7 | C7×M4(2) | C2×C28 | C2×C28 | C22×C14 | C2×M4(2) | M4(2) | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 3 | 6 | 3 | 8 | 6 | 12 | 12 | 6 | 12 |
Matrix representation of C28.3C42 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 |
| 0 | 18 | 15 | 0 | 0 |
| 0 | 0 | 0 | 112 | 79 |
| 0 | 0 | 0 | 34 | 25 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 98 | 88 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 34 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 36 | 0 | 0 |
| 0 | 100 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,98,18,0,0,0,0,15,0,0,0,0,0,112,34,0,0,0,79,25],[98,0,0,0,0,0,98,0,0,0,0,88,15,0,0,0,0,0,112,34,0,0,0,0,1],[1,0,0,0,0,0,112,100,0,0,0,36,1,0,0,0,0,0,112,0,0,0,0,0,112] >;
C28.3C42 in GAP, Magma, Sage, TeX
C_{28}._3C_4^2 % in TeX
G:=Group("C28.3C4^2"); // GroupNames label
G:=SmallGroup(448,112);
// by ID
G=gap.SmallGroup(448,112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,136,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=1,c^4=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b>;
// generators/relations