metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊16D14, C14.182+ 1+4, C4⋊C4⋊49D14, (C4×D4)⋊18D7, (D4×C28)⋊20C2, (C22×C4)⋊5D14, (C4×C28)⋊32C22, D14⋊Q8⋊8C2, C22⋊C4⋊48D14, C4⋊Dic7⋊9C22, C23⋊D14.5C2, D14.D4⋊7C2, (C2×D4).217D14, C42⋊2D7⋊16C2, C42⋊D7⋊32C2, D14.17(C4○D4), C28.48D4⋊11C2, C23.D7⋊9C22, (C2×C14).100C24, (C2×C28).699C23, Dic7⋊C4⋊42C22, D14⋊C4.85C22, (C22×C28)⋊37C22, Dic7.D4⋊7C2, (C4×Dic7)⋊52C22, (C2×Dic14)⋊6C22, C2.19(D4⋊6D14), C7⋊3(C22.45C24), (D4×C14).307C22, C22.12(C4○D28), (C22×D7).35C23, (C23×D7).41C22, C23.174(C22×D7), C22.125(C23×D7), C23.11D14⋊29C2, C23.18D14⋊18C2, C23.23D14⋊16C2, (C22×C14).170C23, (C2×Dic7).207C23, (C22×Dic7).98C22, C4⋊C4⋊D7⋊7C2, (C4×C7⋊D4)⋊43C2, C2.23(D7×C4○D4), (C2×D14⋊C4)⋊22C2, (C7×C4⋊C4)⋊61C22, (D7×C22⋊C4)⋊29C2, C2.49(C2×C4○D28), C14.140(C2×C4○D4), (C2×C4×D7).201C22, (C2×C14).16(C4○D4), (C7×C22⋊C4)⋊57C22, (C2×C4).284(C22×D7), (C2×C7⋊D4).16C22, SmallGroup(448,1009)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊16D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1204 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.45C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C42⋊D7, C42⋊2D7, C23.11D14, D7×C22⋊C4, D14.D4, Dic7.D4, D14⋊Q8, C4⋊C4⋊D7, C28.48D4, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C23.18D14, C23⋊D14, D4×C28, C42⋊16D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, C4○D28, C23×D7, C2×C4○D28, D4⋊6D14, D7×C4○D4, C42⋊16D14
►On 112 points
(1 71 24 90)(2 79 25 98)(3 73 26 92)(4 81 27 86)(5 75 28 94)(6 83 22 88)(7 77 23 96)(8 91 15 72)(9 85 16 80)(10 93 17 74)(11 87 18 82)(12 95 19 76)(13 89 20 84)(14 97 21 78)(29 59 50 111)(30 67 51 105)(31 61 52 99)(32 69 53 107)(33 63 54 101)(34 57 55 109)(35 65 56 103)(36 104 43 66)(37 112 44 60)(38 106 45 68)(39 100 46 62)(40 108 47 70)(41 102 48 64)(42 110 49 58)
(1 38 14 31)(2 39 8 32)(3 40 9 33)(4 41 10 34)(5 42 11 35)(6 36 12 29)(7 37 13 30)(15 53 25 46)(16 54 26 47)(17 55 27 48)(18 56 28 49)(19 50 22 43)(20 51 23 44)(21 52 24 45)(57 81 102 93)(58 82 103 94)(59 83 104 95)(60 84 105 96)(61 71 106 97)(62 72 107 98)(63 73 108 85)(64 74 109 86)(65 75 110 87)(66 76 111 88)(67 77 112 89)(68 78 99 90)(69 79 100 91)(70 80 101 92)
(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 9)(2 8)(3 14)(4 13)(5 12)(6 11)(7 10)(15 25)(16 24)(17 23)(18 22)(19 28)(20 27)(21 26)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(99 101)(102 112)(103 111)(104 110)(105 109)(106 108)
G:=sub<Sym(112)| (1,71,24,90)(2,79,25,98)(3,73,26,92)(4,81,27,86)(5,75,28,94)(6,83,22,88)(7,77,23,96)(8,91,15,72)(9,85,16,80)(10,93,17,74)(11,87,18,82)(12,95,19,76)(13,89,20,84)(14,97,21,78)(29,59,50,111)(30,67,51,105)(31,61,52,99)(32,69,53,107)(33,63,54,101)(34,57,55,109)(35,65,56,103)(36,104,43,66)(37,112,44,60)(38,106,45,68)(39,100,46,62)(40,108,47,70)(41,102,48,64)(42,110,49,58), (1,38,14,31)(2,39,8,32)(3,40,9,33)(4,41,10,34)(5,42,11,35)(6,36,12,29)(7,37,13,30)(15,53,25,46)(16,54,26,47)(17,55,27,48)(18,56,28,49)(19,50,22,43)(20,51,23,44)(21,52,24,45)(57,81,102,93)(58,82,103,94)(59,83,104,95)(60,84,105,96)(61,71,106,97)(62,72,107,98)(63,73,108,85)(64,74,109,86)(65,75,110,87)(66,76,111,88)(67,77,112,89)(68,78,99,90)(69,79,100,91)(70,80,101,92), (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), (1,9)(2,8)(3,14)(4,13)(5,12)(6,11)(7,10)(15,25)(16,24)(17,23)(18,22)(19,28)(20,27)(21,26)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(99,101)(102,112)(103,111)(104,110)(105,109)(106,108)>;
G:=Group( (1,71,24,90)(2,79,25,98)(3,73,26,92)(4,81,27,86)(5,75,28,94)(6,83,22,88)(7,77,23,96)(8,91,15,72)(9,85,16,80)(10,93,17,74)(11,87,18,82)(12,95,19,76)(13,89,20,84)(14,97,21,78)(29,59,50,111)(30,67,51,105)(31,61,52,99)(32,69,53,107)(33,63,54,101)(34,57,55,109)(35,65,56,103)(36,104,43,66)(37,112,44,60)(38,106,45,68)(39,100,46,62)(40,108,47,70)(41,102,48,64)(42,110,49,58), (1,38,14,31)(2,39,8,32)(3,40,9,33)(4,41,10,34)(5,42,11,35)(6,36,12,29)(7,37,13,30)(15,53,25,46)(16,54,26,47)(17,55,27,48)(18,56,28,49)(19,50,22,43)(20,51,23,44)(21,52,24,45)(57,81,102,93)(58,82,103,94)(59,83,104,95)(60,84,105,96)(61,71,106,97)(62,72,107,98)(63,73,108,85)(64,74,109,86)(65,75,110,87)(66,76,111,88)(67,77,112,89)(68,78,99,90)(69,79,100,91)(70,80,101,92), (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), (1,9)(2,8)(3,14)(4,13)(5,12)(6,11)(7,10)(15,25)(16,24)(17,23)(18,22)(19,28)(20,27)(21,26)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(99,101)(102,112)(103,111)(104,110)(105,109)(106,108) );
G=PermutationGroup([[(1,71,24,90),(2,79,25,98),(3,73,26,92),(4,81,27,86),(5,75,28,94),(6,83,22,88),(7,77,23,96),(8,91,15,72),(9,85,16,80),(10,93,17,74),(11,87,18,82),(12,95,19,76),(13,89,20,84),(14,97,21,78),(29,59,50,111),(30,67,51,105),(31,61,52,99),(32,69,53,107),(33,63,54,101),(34,57,55,109),(35,65,56,103),(36,104,43,66),(37,112,44,60),(38,106,45,68),(39,100,46,62),(40,108,47,70),(41,102,48,64),(42,110,49,58)], [(1,38,14,31),(2,39,8,32),(3,40,9,33),(4,41,10,34),(5,42,11,35),(6,36,12,29),(7,37,13,30),(15,53,25,46),(16,54,26,47),(17,55,27,48),(18,56,28,49),(19,50,22,43),(20,51,23,44),(21,52,24,45),(57,81,102,93),(58,82,103,94),(59,83,104,95),(60,84,105,96),(61,71,106,97),(62,72,107,98),(63,73,108,85),(64,74,109,86),(65,75,110,87),(66,76,111,88),(67,77,112,89),(68,78,99,90),(69,79,100,91),(70,80,101,92)], [(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)], [(1,9),(2,8),(3,14),(4,13),(5,12),(6,11),(7,10),(15,25),(16,24),(17,23),(18,22),(19,28),(20,27),(21,26),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(57,67),(58,66),(59,65),(60,64),(61,63),(68,70),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(99,101),(102,112),(103,111),(104,110),(105,109),(106,108)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 28 | 2 | ··· | 2 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4⋊6D14 | D7×C4○D4 |
| kernel | C42⋊16D14 | C42⋊D7 | C42⋊2D7 | C23.11D14 | D7×C22⋊C4 | D14.D4 | Dic7.D4 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C2×D14⋊C4 | C4×C7⋊D4 | C23.23D14 | C23.18D14 | C23⋊D14 | D4×C28 | C4×D4 | D14 | C2×C14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of C42⋊16D14 ►in GL6(𝔽29)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 0 | 7 | 20 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 12 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 10 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,7,0,0,0,0,1,20],[2,12,0,0,0,0,2,27,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,19,0,0,0,0,10,10,0,0,0,0,0,0,1,11,0,0,0,0,0,28],[28,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,18,0,0,0,0,0,1] >;
C42⋊16D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{16}D_{14} % in TeX
G:=Group("C4^2:16D14"); // GroupNames label
G:=SmallGroup(448,1009);
// by ID
G=gap.SmallGroup(448,1009);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations