metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D14, C14.1262+ 1+4, (C2×Q8)⋊8D14, (C4×D28)⋊45C2, (C4×C28)⋊24C22, C22⋊C4⋊34D14, C4.4D4⋊12D7, C42⋊2D7⋊8C2, C23⋊D14⋊24C2, D14⋊D4⋊42C2, C22⋊D28⋊25C2, D14⋊C4⋊69C22, D14⋊3Q8⋊30C2, D14.8(C4○D4), (C2×D4).110D14, C4⋊Dic7⋊41C22, (Q8×C14)⋊14C22, Dic7⋊4D4⋊31C2, Dic7⋊D4⋊34C2, D14.D4⋊43C2, C28.23D4⋊22C2, (C2×C14).222C24, (C2×C28).631C23, Dic7⋊C4⋊36C22, C7⋊8(C22.32C24), (C4×Dic7)⋊36C22, C2.50(D4⋊8D14), C2.75(D4⋊6D14), C23.44(C22×D7), (D4×C14).210C22, (C2×D28).224C22, C22.D28⋊25C2, C23.D14⋊39C2, (C22×C14).52C23, (C23×D7).65C22, (C22×D7).96C23, C22.243(C23×D7), C23.D7.56C22, (C2×Dic7).254C23, (C22×Dic7)⋊27C22, C2.78(D7×C4○D4), (C2×C4×D7)⋊52C22, (D7×C22⋊C4)⋊18C2, C14.189(C2×C4○D4), (C7×C4.4D4)⋊14C2, (C2×C7⋊D4)⋊24C22, (C7×C22⋊C4)⋊30C22, (C2×C4).197(C22×D7), SmallGroup(448,1131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊20D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 1388 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C4×D28, C42⋊2D7, C23.D14, D7×C22⋊C4, Dic7⋊4D4, C22⋊D28, D14.D4, D14⋊D4, C22.D28, C23⋊D14, Dic7⋊D4, D14⋊3Q8, C28.23D4, C7×C4.4D4, C42⋊20D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C23×D7, D4⋊6D14, D7×C4○D4, D4⋊8D14, C42⋊20D14
►On 112 points
Generators in S112
(1 65 12 94)(2 59 13 88)(3 67 14 96)(4 61 8 90)(5 69 9 98)(6 63 10 92)(7 57 11 86)(15 70 24 85)(16 64 25 93)(17 58 26 87)(18 66 27 95)(19 60 28 89)(20 68 22 97)(21 62 23 91)(29 100 50 107)(30 80 51 73)(31 102 52 109)(32 82 53 75)(33 104 54 111)(34 84 55 77)(35 106 56 99)(36 72 43 79)(37 108 44 101)(38 74 45 81)(39 110 46 103)(40 76 47 83)(41 112 48 105)(42 78 49 71)
(1 108 17 73)(2 102 18 81)(3 110 19 75)(4 104 20 83)(5 112 21 77)(6 106 15 71)(7 100 16 79)(8 111 22 76)(9 105 23 84)(10 99 24 78)(11 107 25 72)(12 101 26 80)(13 109 27 74)(14 103 28 82)(29 93 43 86)(30 65 44 58)(31 95 45 88)(32 67 46 60)(33 97 47 90)(34 69 48 62)(35 85 49 92)(36 57 50 64)(37 87 51 94)(38 59 52 66)(39 89 53 96)(40 61 54 68)(41 91 55 98)(42 63 56 70)
(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 11)(2 10)(3 9)(4 8)(5 14)(6 13)(7 12)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(21 28)(29 37)(30 36)(31 35)(32 34)(38 42)(39 41)(43 51)(44 50)(45 49)(46 48)(52 56)(53 55)(57 87)(58 86)(59 85)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 102)(72 101)(73 100)(74 99)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 106)(82 105)(83 104)(84 103)
G:=sub<Sym(112)| (1,65,12,94)(2,59,13,88)(3,67,14,96)(4,61,8,90)(5,69,9,98)(6,63,10,92)(7,57,11,86)(15,70,24,85)(16,64,25,93)(17,58,26,87)(18,66,27,95)(19,60,28,89)(20,68,22,97)(21,62,23,91)(29,100,50,107)(30,80,51,73)(31,102,52,109)(32,82,53,75)(33,104,54,111)(34,84,55,77)(35,106,56,99)(36,72,43,79)(37,108,44,101)(38,74,45,81)(39,110,46,103)(40,76,47,83)(41,112,48,105)(42,78,49,71), (1,108,17,73)(2,102,18,81)(3,110,19,75)(4,104,20,83)(5,112,21,77)(6,106,15,71)(7,100,16,79)(8,111,22,76)(9,105,23,84)(10,99,24,78)(11,107,25,72)(12,101,26,80)(13,109,27,74)(14,103,28,82)(29,93,43,86)(30,65,44,58)(31,95,45,88)(32,67,46,60)(33,97,47,90)(34,69,48,62)(35,85,49,92)(36,57,50,64)(37,87,51,94)(38,59,52,66)(39,89,53,96)(40,61,54,68)(41,91,55,98)(42,63,56,70), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(29,37)(30,36)(31,35)(32,34)(38,42)(39,41)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55)(57,87)(58,86)(59,85)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,102)(72,101)(73,100)(74,99)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)>;
G:=Group( (1,65,12,94)(2,59,13,88)(3,67,14,96)(4,61,8,90)(5,69,9,98)(6,63,10,92)(7,57,11,86)(15,70,24,85)(16,64,25,93)(17,58,26,87)(18,66,27,95)(19,60,28,89)(20,68,22,97)(21,62,23,91)(29,100,50,107)(30,80,51,73)(31,102,52,109)(32,82,53,75)(33,104,54,111)(34,84,55,77)(35,106,56,99)(36,72,43,79)(37,108,44,101)(38,74,45,81)(39,110,46,103)(40,76,47,83)(41,112,48,105)(42,78,49,71), (1,108,17,73)(2,102,18,81)(3,110,19,75)(4,104,20,83)(5,112,21,77)(6,106,15,71)(7,100,16,79)(8,111,22,76)(9,105,23,84)(10,99,24,78)(11,107,25,72)(12,101,26,80)(13,109,27,74)(14,103,28,82)(29,93,43,86)(30,65,44,58)(31,95,45,88)(32,67,46,60)(33,97,47,90)(34,69,48,62)(35,85,49,92)(36,57,50,64)(37,87,51,94)(38,59,52,66)(39,89,53,96)(40,61,54,68)(41,91,55,98)(42,63,56,70), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(29,37)(30,36)(31,35)(32,34)(38,42)(39,41)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55)(57,87)(58,86)(59,85)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,102)(72,101)(73,100)(74,99)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103) );
G=PermutationGroup([[(1,65,12,94),(2,59,13,88),(3,67,14,96),(4,61,8,90),(5,69,9,98),(6,63,10,92),(7,57,11,86),(15,70,24,85),(16,64,25,93),(17,58,26,87),(18,66,27,95),(19,60,28,89),(20,68,22,97),(21,62,23,91),(29,100,50,107),(30,80,51,73),(31,102,52,109),(32,82,53,75),(33,104,54,111),(34,84,55,77),(35,106,56,99),(36,72,43,79),(37,108,44,101),(38,74,45,81),(39,110,46,103),(40,76,47,83),(41,112,48,105),(42,78,49,71)], [(1,108,17,73),(2,102,18,81),(3,110,19,75),(4,104,20,83),(5,112,21,77),(6,106,15,71),(7,100,16,79),(8,111,22,76),(9,105,23,84),(10,99,24,78),(11,107,25,72),(12,101,26,80),(13,109,27,74),(14,103,28,82),(29,93,43,86),(30,65,44,58),(31,95,45,88),(32,67,46,60),(33,97,47,90),(34,69,48,62),(35,85,49,92),(36,57,50,64),(37,87,51,94),(38,59,52,66),(39,89,53,96),(40,61,54,68),(41,91,55,98),(42,63,56,70)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,14),(6,13),(7,12),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(21,28),(29,37),(30,36),(31,35),(32,34),(38,42),(39,41),(43,51),(44,50),(45,49),(46,48),(52,56),(53,55),(57,87),(58,86),(59,85),(60,98),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,102),(72,101),(73,100),(74,99),(75,112),(76,111),(77,110),(78,109),(79,108),(80,107),(81,106),(82,105),(83,104),(84,103)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 | D7×C4○D4 | D4⋊8D14 |
| kernel | C42⋊20D14 | C4×D28 | C42⋊2D7 | C23.D14 | D7×C22⋊C4 | Dic7⋊4D4 | C22⋊D28 | D14.D4 | D14⋊D4 | C22.D28 | C23⋊D14 | Dic7⋊D4 | D14⋊3Q8 | C28.23D4 | C7×C4.4D4 | C4.4D4 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 12 | 3 | 3 | 2 | 6 | 6 | 6 |
Matrix representation of C42⋊20D14 ►in GL8(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 23 | 2 |
| 0 | 0 | 0 | 0 | 0 | 17 | 25 | 2 |
| 0 | 0 | 0 | 0 | 28 | 1 | 12 | 0 |
| 0 | 0 | 0 | 0 | 27 | 3 | 0 | 12 |
| 1 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 16 |
| 20 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 3 | 8 | 8 | 0 | 0 | 0 | 0 |
| 22 | 26 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 | 0 | 8 |
| 0 | 0 | 0 | 0 | 16 | 26 | 18 | 11 |
| 26 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 26 | 21 | 21 | 0 | 0 | 0 | 0 |
| 17 | 17 | 26 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 18 | 18 | 8 |
| 0 | 0 | 0 | 0 | 26 | 16 | 14 | 11 |
G:=sub<GL(8,GF(29))| [17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,28,27,0,0,0,0,0,17,1,3,0,0,0,0,23,25,12,0,0,0,0,0,2,2,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,28,0,0,0,0,0,14,4,0,28,0,0,0,0,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,0,0,0,0,13,26,0,0,0,0,0,0,18,16],[20,16,21,22,0,0,0,0,21,27,3,26,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,21,8,18,16,0,0,0,0,21,26,2,26,0,0,0,0,0,0,0,18,0,0,0,0,0,0,8,11],[26,22,8,17,0,0,0,0,26,3,26,17,0,0,0,0,0,0,21,26,0,0,0,0,0,0,21,8,0,0,0,0,0,0,0,0,21,26,2,26,0,0,0,0,21,8,18,16,0,0,0,0,0,0,18,14,0,0,0,0,0,0,8,11] >;
C42⋊20D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_{14} % in TeX
G:=Group("C4^2:20D14"); // GroupNames label
G:=SmallGroup(448,1131);
// by ID
G=gap.SmallGroup(448,1131);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations