metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊8D14, C4⋊C4⋊42D14, (C4×D28)⋊7C2, (C2×C28)⋊10D4, (C2×C4)⋊11D28, (C4×C28)⋊5C22, C4.70(C2×D28), D14⋊1(C4○D4), C4⋊D28⋊42C2, C28.223(C2×D4), C42⋊C2⋊8D7, C22⋊D28⋊29C2, D14⋊C4⋊51C22, D14⋊2Q8⋊47C2, (C2×D28)⋊52C22, (C2×C14).68C24, C4⋊Dic7⋊55C22, C22⋊C4.92D14, C14.12(C22×D4), C2.14(C22×D28), C22.19(C2×D28), (C2×C28).143C23, C7⋊1(C22.19C24), (C22×C4).365D14, C22.97(C23×D7), (C2×Dic14)⋊61C22, C22.D28⋊32C2, (C22×D7).18C23, C23.156(C22×D7), (C22×C28).228C22, (C22×C14).138C23, (C2×Dic7).197C23, (C23×D7).104C22, (C22×Dic7).217C22, C2.9(D7×C4○D4), (D7×C22×C4)⋊2C2, (C2×C4×D7)⋊44C22, (C2×C4○D28)⋊17C2, (C7×C4⋊C4)⋊52C22, (C2×C14).49(C2×D4), C14.133(C2×C4○D4), (C7×C42⋊C2)⋊10C2, (C2×C4).575(C22×D7), (C2×C7⋊D4).99C22, (C7×C22⋊C4).100C22, SmallGroup(448,977)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊8D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1716 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22.19C24, C4⋊Dic7, D14⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, C23×D7, C4×D28, C22⋊D28, C22.D28, C4⋊D28, D14⋊2Q8, C7×C42⋊C2, D7×C22×C4, C2×C4○D28, C42⋊8D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, D28, C22×D7, C22.19C24, C2×D28, C23×D7, C22×D28, D7×C4○D4, C42⋊8D14
►On 112 points
(1 104 36 58)(2 112 37 66)(3 106 38 60)(4 100 39 68)(5 108 40 62)(6 102 41 70)(7 110 42 64)(8 98 15 71)(9 92 16 79)(10 86 17 73)(11 94 18 81)(12 88 19 75)(13 96 20 83)(14 90 21 77)(22 78 52 91)(23 72 53 85)(24 80 54 93)(25 74 55 87)(26 82 56 95)(27 76 50 89)(28 84 51 97)(29 105 44 59)(30 99 45 67)(31 107 46 61)(32 101 47 69)(33 109 48 63)(34 103 49 57)(35 111 43 65)
(1 51 35 14)(2 52 29 8)(3 53 30 9)(4 54 31 10)(5 55 32 11)(6 56 33 12)(7 50 34 13)(15 37 22 44)(16 38 23 45)(17 39 24 46)(18 40 25 47)(19 41 26 48)(20 42 27 49)(21 36 28 43)(57 83 64 76)(58 84 65 77)(59 71 66 78)(60 72 67 79)(61 73 68 80)(62 74 69 81)(63 75 70 82)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(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 30)(2 29)(3 35)(4 34)(5 33)(6 32)(7 31)(8 52)(9 51)(10 50)(11 56)(12 55)(13 54)(14 53)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 100)(58 99)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)
G:=sub<Sym(112)| (1,104,36,58)(2,112,37,66)(3,106,38,60)(4,100,39,68)(5,108,40,62)(6,102,41,70)(7,110,42,64)(8,98,15,71)(9,92,16,79)(10,86,17,73)(11,94,18,81)(12,88,19,75)(13,96,20,83)(14,90,21,77)(22,78,52,91)(23,72,53,85)(24,80,54,93)(25,74,55,87)(26,82,56,95)(27,76,50,89)(28,84,51,97)(29,105,44,59)(30,99,45,67)(31,107,46,61)(32,101,47,69)(33,109,48,63)(34,103,49,57)(35,111,43,65), (1,51,35,14)(2,52,29,8)(3,53,30,9)(4,54,31,10)(5,55,32,11)(6,56,33,12)(7,50,34,13)(15,37,22,44)(16,38,23,45)(17,39,24,46)(18,40,25,47)(19,41,26,48)(20,42,27,49)(21,36,28,43)(57,83,64,76)(58,84,65,77)(59,71,66,78)(60,72,67,79)(61,73,68,80)(62,74,69,81)(63,75,70,82)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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), (1,30)(2,29)(3,35)(4,34)(5,33)(6,32)(7,31)(8,52)(9,51)(10,50)(11,56)(12,55)(13,54)(14,53)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,100)(58,99)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)>;
G:=Group( (1,104,36,58)(2,112,37,66)(3,106,38,60)(4,100,39,68)(5,108,40,62)(6,102,41,70)(7,110,42,64)(8,98,15,71)(9,92,16,79)(10,86,17,73)(11,94,18,81)(12,88,19,75)(13,96,20,83)(14,90,21,77)(22,78,52,91)(23,72,53,85)(24,80,54,93)(25,74,55,87)(26,82,56,95)(27,76,50,89)(28,84,51,97)(29,105,44,59)(30,99,45,67)(31,107,46,61)(32,101,47,69)(33,109,48,63)(34,103,49,57)(35,111,43,65), (1,51,35,14)(2,52,29,8)(3,53,30,9)(4,54,31,10)(5,55,32,11)(6,56,33,12)(7,50,34,13)(15,37,22,44)(16,38,23,45)(17,39,24,46)(18,40,25,47)(19,41,26,48)(20,42,27,49)(21,36,28,43)(57,83,64,76)(58,84,65,77)(59,71,66,78)(60,72,67,79)(61,73,68,80)(62,74,69,81)(63,75,70,82)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (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), (1,30)(2,29)(3,35)(4,34)(5,33)(6,32)(7,31)(8,52)(9,51)(10,50)(11,56)(12,55)(13,54)(14,53)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,100)(58,99)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92) );
G=PermutationGroup([[(1,104,36,58),(2,112,37,66),(3,106,38,60),(4,100,39,68),(5,108,40,62),(6,102,41,70),(7,110,42,64),(8,98,15,71),(9,92,16,79),(10,86,17,73),(11,94,18,81),(12,88,19,75),(13,96,20,83),(14,90,21,77),(22,78,52,91),(23,72,53,85),(24,80,54,93),(25,74,55,87),(26,82,56,95),(27,76,50,89),(28,84,51,97),(29,105,44,59),(30,99,45,67),(31,107,46,61),(32,101,47,69),(33,109,48,63),(34,103,49,57),(35,111,43,65)], [(1,51,35,14),(2,52,29,8),(3,53,30,9),(4,54,31,10),(5,55,32,11),(6,56,33,12),(7,50,34,13),(15,37,22,44),(16,38,23,45),(17,39,24,46),(18,40,25,47),(19,41,26,48),(20,42,27,49),(21,36,28,43),(57,83,64,76),(58,84,65,77),(59,71,66,78),(60,72,67,79),(61,73,68,80),(62,74,69,81),(63,75,70,82),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(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)], [(1,30),(2,29),(3,35),(4,34),(5,33),(6,32),(7,31),(8,52),(9,51),(10,50),(11,56),(12,55),(13,54),(14,53),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46),(57,100),(58,99),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D28 | D7×C4○D4 |
| kernel | C42⋊8D14 | C4×D28 | C22⋊D28 | C22.D28 | C4⋊D28 | D14⋊2Q8 | C7×C42⋊C2 | D7×C22×C4 | C2×C4○D28 | C2×C28 | C42⋊C2 | D14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 8 | 6 | 6 | 6 | 3 | 24 | 12 |
Matrix representation of C42⋊8D14 ►in GL4(𝔽29) generated by
| 2 | 18 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 21 | 9 |
| 0 | 0 | 22 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 21 | 21 | 0 | 0 |
| 8 | 26 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 28 |
| 21 | 21 | 0 | 0 |
| 26 | 8 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [2,11,0,0,18,27,0,0,0,0,21,22,0,0,9,8],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[21,8,0,0,21,26,0,0,0,0,1,0,0,0,6,28],[21,26,0,0,21,8,0,0,0,0,28,0,0,0,0,28] >;
C42⋊8D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8D_{14} % in TeX
G:=Group("C4^2:8D14"); // GroupNames label
G:=SmallGroup(448,977);
// by ID
G=gap.SmallGroup(448,977);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations