metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊4D14, C4○D28⋊5C4, (C2×D28)⋊12C4, (C4×C28)⋊3C22, C4.83(C2×D28), Dic14⋊C4⋊2C2, D28.22(C2×C4), (C2×C4).147D28, (C2×C28).144D4, C28.303(C2×D4), C42⋊C2⋊4D7, C4.10(D14⋊C4), (C2×Dic14)⋊12C4, (C22×C14).78D4, C28.23(C22⋊C4), (C2×C28).794C23, C28.110(C22×C4), Dic14.23(C2×C4), C7⋊2(C42⋊C22), C4○D28.38C22, (C22×C4).113D14, C23.21(C7⋊D4), C4.Dic7⋊20C22, C22.25(D14⋊C4), (C22×C28).154C22, C4.68(C2×C4×D7), (C2×C4).46(C4×D7), (C2×C28).94(C2×C4), C2.20(C2×D14⋊C4), (C2×C4○D28).8C2, (C7×C42⋊C2)⋊4C2, (C2×C14).461(C2×D4), (C2×C4).45(C7⋊D4), C14.47(C2×C22⋊C4), (C2×C4.Dic7)⋊10C2, C22.27(C2×C7⋊D4), (C2×C4).708(C22×D7), (C2×C14).17(C22⋊C4), SmallGroup(448,539)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊4D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 756 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C42⋊C22, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, Dic14⋊C4, C2×C4.Dic7, C7×C42⋊C2, C2×C4○D28, C42⋊4D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C42⋊C22, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2×D14⋊C4, C42⋊4D14
►On 112 points
(1 21)(3 23)(5 25)(7 27)(9 15)(11 17)(13 19)(29 92 59 56)(30 43 60 93)(31 94 61 44)(32 45 62 95)(33 96 63 46)(34 47 64 97)(35 98 65 48)(36 49 66 85)(37 86 67 50)(38 51 68 87)(39 88 69 52)(40 53 70 89)(41 90 57 54)(42 55 58 91)(71 109)(73 111)(75 99)(77 101)(79 103)(81 105)(83 107)
(1 109 21 71)(2 110 22 72)(3 111 23 73)(4 112 24 74)(5 99 25 75)(6 100 26 76)(7 101 27 77)(8 102 28 78)(9 103 15 79)(10 104 16 80)(11 105 17 81)(12 106 18 82)(13 107 19 83)(14 108 20 84)(29 92 59 56)(30 93 60 43)(31 94 61 44)(32 95 62 45)(33 96 63 46)(34 97 64 47)(35 98 65 48)(36 85 66 49)(37 86 67 50)(38 87 68 51)(39 88 69 52)(40 89 70 53)(41 90 57 54)(42 91 58 55)
(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 32)(2 31)(3 30)(4 29)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 70)(28 69)(43 111)(44 110)(45 109)(46 108)(47 107)(48 106)(49 105)(50 104)(51 103)(52 102)(53 101)(54 100)(55 99)(56 112)(71 95)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 98)(83 97)(84 96)
G:=sub<Sym(112)| (1,21)(3,23)(5,25)(7,27)(9,15)(11,17)(13,19)(29,92,59,56)(30,43,60,93)(31,94,61,44)(32,45,62,95)(33,96,63,46)(34,47,64,97)(35,98,65,48)(36,49,66,85)(37,86,67,50)(38,51,68,87)(39,88,69,52)(40,53,70,89)(41,90,57,54)(42,55,58,91)(71,109)(73,111)(75,99)(77,101)(79,103)(81,105)(83,107), (1,109,21,71)(2,110,22,72)(3,111,23,73)(4,112,24,74)(5,99,25,75)(6,100,26,76)(7,101,27,77)(8,102,28,78)(9,103,15,79)(10,104,16,80)(11,105,17,81)(12,106,18,82)(13,107,19,83)(14,108,20,84)(29,92,59,56)(30,93,60,43)(31,94,61,44)(32,95,62,45)(33,96,63,46)(34,97,64,47)(35,98,65,48)(36,85,66,49)(37,86,67,50)(38,87,68,51)(39,88,69,52)(40,89,70,53)(41,90,57,54)(42,91,58,55), (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), (1,32)(2,31)(3,30)(4,29)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,70)(28,69)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,105)(50,104)(51,103)(52,102)(53,101)(54,100)(55,99)(56,112)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,98)(83,97)(84,96)>;
G:=Group( (1,21)(3,23)(5,25)(7,27)(9,15)(11,17)(13,19)(29,92,59,56)(30,43,60,93)(31,94,61,44)(32,45,62,95)(33,96,63,46)(34,47,64,97)(35,98,65,48)(36,49,66,85)(37,86,67,50)(38,51,68,87)(39,88,69,52)(40,53,70,89)(41,90,57,54)(42,55,58,91)(71,109)(73,111)(75,99)(77,101)(79,103)(81,105)(83,107), (1,109,21,71)(2,110,22,72)(3,111,23,73)(4,112,24,74)(5,99,25,75)(6,100,26,76)(7,101,27,77)(8,102,28,78)(9,103,15,79)(10,104,16,80)(11,105,17,81)(12,106,18,82)(13,107,19,83)(14,108,20,84)(29,92,59,56)(30,93,60,43)(31,94,61,44)(32,95,62,45)(33,96,63,46)(34,97,64,47)(35,98,65,48)(36,85,66,49)(37,86,67,50)(38,87,68,51)(39,88,69,52)(40,89,70,53)(41,90,57,54)(42,91,58,55), (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), (1,32)(2,31)(3,30)(4,29)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,70)(28,69)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,105)(50,104)(51,103)(52,102)(53,101)(54,100)(55,99)(56,112)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,98)(83,97)(84,96) );
G=PermutationGroup([[(1,21),(3,23),(5,25),(7,27),(9,15),(11,17),(13,19),(29,92,59,56),(30,43,60,93),(31,94,61,44),(32,45,62,95),(33,96,63,46),(34,47,64,97),(35,98,65,48),(36,49,66,85),(37,86,67,50),(38,51,68,87),(39,88,69,52),(40,53,70,89),(41,90,57,54),(42,55,58,91),(71,109),(73,111),(75,99),(77,101),(79,103),(81,105),(83,107)], [(1,109,21,71),(2,110,22,72),(3,111,23,73),(4,112,24,74),(5,99,25,75),(6,100,26,76),(7,101,27,77),(8,102,28,78),(9,103,15,79),(10,104,16,80),(11,105,17,81),(12,106,18,82),(13,107,19,83),(14,108,20,84),(29,92,59,56),(30,93,60,43),(31,94,61,44),(32,95,62,45),(33,96,63,46),(34,97,64,47),(35,98,65,48),(36,85,66,49),(37,86,67,50),(38,87,68,51),(39,88,69,52),(40,89,70,53),(41,90,57,54),(42,91,58,55)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,70),(28,69),(43,111),(44,110),(45,109),(46,108),(47,107),(48,106),(49,105),(50,104),(51,103),(52,102),(53,101),(54,100),(55,99),(56,112),(71,95),(72,94),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,98),(83,97),(84,96)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C7⋊D4 | C42⋊C22 | C42⋊4D14 |
| kernel | C42⋊4D14 | Dic14⋊C4 | C2×C4.Dic7 | C7×C42⋊C2 | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C28 | C22×C14 | C42⋊C2 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 3 | 6 | 3 | 12 | 12 | 6 | 6 | 2 | 12 |
Matrix representation of C42⋊4D14 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 98 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 64 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 83 |
| 0 | 0 | 83 | 0 |
| 0 | 0 | 0 | 83 |
| 0 | 0 | 83 | 0 |
| 0 | 64 | 0 | 0 |
| 64 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,15,0,0,0,0,98],[98,0,0,0,0,98,0,0,0,0,15,0,0,0,0,15],[0,64,0,0,64,0,0,0,0,0,0,83,0,0,83,0],[0,0,0,64,0,0,64,0,0,83,0,0,83,0,0,0] >;
C42⋊4D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4D_{14} % in TeX
G:=Group("C4^2:4D14"); // GroupNames label
G:=SmallGroup(448,539);
// by ID
G=gap.SmallGroup(448,539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,387,58,1123,1684,102,18822]);
// 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*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations