metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D14, C14.802+ 1+4, C4⋊1D4⋊9D7, (C2×D4)⋊13D14, (C4×C28)⋊37C22, C23⋊D14⋊28C2, (D4×C14)⋊34C22, C42⋊2D7⋊19C2, Dic7⋊D4⋊39C2, (C2×C28).638C23, (C2×C14).264C24, Dic7⋊C4⋊37C22, D14⋊C4.75C22, C23.D7⋊38C22, C2.84(D4⋊6D14), C23.70(C22×D7), C7⋊5(C22.54C24), (C22×C14).78C23, (C23×D7).73C22, C22.285(C23×D7), C23.18D14⋊28C2, (C2×Dic7).138C23, (C22×Dic7)⋊30C22, (C22×D7).118C23, (C7×C4⋊1D4)⋊15C2, (C2×C4).216(C22×D7), (C2×C7⋊D4).80C22, SmallGroup(448,1173)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊28D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1356 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22.54C24, Dic7⋊C4, D14⋊C4, C23.D7, C4×C28, C22×Dic7, C2×C7⋊D4, D4×C14, C23×D7, C42⋊2D7, C23.18D14, C23⋊D14, Dic7⋊D4, C7×C4⋊1D4, C42⋊28D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C22.54C24, C23×D7, D4⋊6D14, C42⋊28D14
►On 112 points
Generators in S112
(1 95 19 88)(2 89 20 96)(3 97 21 90)(4 91 15 98)(5 85 16 92)(6 93 17 86)(7 87 18 94)(8 84 22 77)(9 78 23 71)(10 72 24 79)(11 80 25 73)(12 74 26 81)(13 82 27 75)(14 76 28 83)(29 52 111 60)(30 61 112 53)(31 54 99 62)(32 63 100 55)(33 56 101 64)(34 65 102 43)(35 44 103 66)(36 67 104 45)(37 46 105 68)(38 69 106 47)(39 48 107 70)(40 57 108 49)(41 50 109 58)(42 59 110 51)
(1 112 12 105)(2 106 13 99)(3 100 14 107)(4 108 8 101)(5 102 9 109)(6 110 10 103)(7 104 11 111)(15 40 22 33)(16 34 23 41)(17 42 24 35)(18 36 25 29)(19 30 26 37)(20 38 27 31)(21 32 28 39)(43 78 58 85)(44 86 59 79)(45 80 60 87)(46 88 61 81)(47 82 62 89)(48 90 63 83)(49 84 64 91)(50 92 65 71)(51 72 66 93)(52 94 67 73)(53 74 68 95)(54 96 69 75)(55 76 70 97)(56 98 57 77)
(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 25)(2 24)(3 23)(4 22)(5 28)(6 27)(7 26)(8 15)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 30)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(111 112)
G:=sub<Sym(112)| (1,95,19,88)(2,89,20,96)(3,97,21,90)(4,91,15,98)(5,85,16,92)(6,93,17,86)(7,87,18,94)(8,84,22,77)(9,78,23,71)(10,72,24,79)(11,80,25,73)(12,74,26,81)(13,82,27,75)(14,76,28,83)(29,52,111,60)(30,61,112,53)(31,54,99,62)(32,63,100,55)(33,56,101,64)(34,65,102,43)(35,44,103,66)(36,67,104,45)(37,46,105,68)(38,69,106,47)(39,48,107,70)(40,57,108,49)(41,50,109,58)(42,59,110,51), (1,112,12,105)(2,106,13,99)(3,100,14,107)(4,108,8,101)(5,102,9,109)(6,110,10,103)(7,104,11,111)(15,40,22,33)(16,34,23,41)(17,42,24,35)(18,36,25,29)(19,30,26,37)(20,38,27,31)(21,32,28,39)(43,78,58,85)(44,86,59,79)(45,80,60,87)(46,88,61,81)(47,82,62,89)(48,90,63,83)(49,84,64,91)(50,92,65,71)(51,72,66,93)(52,94,67,73)(53,74,68,95)(54,96,69,75)(55,76,70,97)(56,98,57,77), (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), (1,25)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,30)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(111,112)>;
G:=Group( (1,95,19,88)(2,89,20,96)(3,97,21,90)(4,91,15,98)(5,85,16,92)(6,93,17,86)(7,87,18,94)(8,84,22,77)(9,78,23,71)(10,72,24,79)(11,80,25,73)(12,74,26,81)(13,82,27,75)(14,76,28,83)(29,52,111,60)(30,61,112,53)(31,54,99,62)(32,63,100,55)(33,56,101,64)(34,65,102,43)(35,44,103,66)(36,67,104,45)(37,46,105,68)(38,69,106,47)(39,48,107,70)(40,57,108,49)(41,50,109,58)(42,59,110,51), (1,112,12,105)(2,106,13,99)(3,100,14,107)(4,108,8,101)(5,102,9,109)(6,110,10,103)(7,104,11,111)(15,40,22,33)(16,34,23,41)(17,42,24,35)(18,36,25,29)(19,30,26,37)(20,38,27,31)(21,32,28,39)(43,78,58,85)(44,86,59,79)(45,80,60,87)(46,88,61,81)(47,82,62,89)(48,90,63,83)(49,84,64,91)(50,92,65,71)(51,72,66,93)(52,94,67,73)(53,74,68,95)(54,96,69,75)(55,76,70,97)(56,98,57,77), (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), (1,25)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,30)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(111,112) );
G=PermutationGroup([[(1,95,19,88),(2,89,20,96),(3,97,21,90),(4,91,15,98),(5,85,16,92),(6,93,17,86),(7,87,18,94),(8,84,22,77),(9,78,23,71),(10,72,24,79),(11,80,25,73),(12,74,26,81),(13,82,27,75),(14,76,28,83),(29,52,111,60),(30,61,112,53),(31,54,99,62),(32,63,100,55),(33,56,101,64),(34,65,102,43),(35,44,103,66),(36,67,104,45),(37,46,105,68),(38,69,106,47),(39,48,107,70),(40,57,108,49),(41,50,109,58),(42,59,110,51)], [(1,112,12,105),(2,106,13,99),(3,100,14,107),(4,108,8,101),(5,102,9,109),(6,110,10,103),(7,104,11,111),(15,40,22,33),(16,34,23,41),(17,42,24,35),(18,36,25,29),(19,30,26,37),(20,38,27,31),(21,32,28,39),(43,78,58,85),(44,86,59,79),(45,80,60,87),(46,88,61,81),(47,82,62,89),(48,90,63,83),(49,84,64,91),(50,92,65,71),(51,72,66,93),(52,94,67,73),(53,74,68,95),(54,96,69,75),(55,76,70,97),(56,98,57,77)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,28),(6,27),(7,26),(8,15),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,30),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(71,83),(72,82),(73,81),(74,80),(75,79),(76,78),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105),(111,112)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | 2+ 1+4 | D4⋊6D14 |
| kernel | C42⋊28D14 | C42⋊2D7 | C23.18D14 | C23⋊D14 | Dic7⋊D4 | C7×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C14 | C2 |
| # reps | 1 | 2 | 3 | 3 | 6 | 1 | 3 | 3 | 18 | 3 | 18 |
Matrix representation of C42⋊28D14 ►in GL8(𝔽29)
| 5 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 1 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 20 |
| 21 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 21 | 8 | 8 | 0 | 0 | 0 | 0 |
| 8 | 26 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 7 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 19 |
G:=sub<GL(8,GF(29))| [5,13,0,0,0,0,0,0,16,24,0,0,0,0,0,0,0,0,5,13,0,0,0,0,0,0,16,24,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,9,15,0,0,0,0,0,0,14,20,0,0,0,0,0,0,0,0,9,15,0,0,0,0,0,0,14,20],[21,8,21,8,0,0,0,0,21,26,21,26,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,10,19,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7],[8,3,0,0,0,0,0,0,8,21,0,0,0,0,0,0,0,0,8,3,0,0,0,0,0,0,8,21,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,10,22,0,0,0,0,0,0,10,19] >;
C42⋊28D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_{14} % in TeX
G:=Group("C4^2:28D14"); // GroupNames label
G:=SmallGroup(448,1173);
// by ID
G=gap.SmallGroup(448,1173);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations