metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1C4≀C2, (C2×D28)⋊1C4, C4⋊Dic7⋊1C4, (C2×C14).29D8, (C2×C28).221D4, (C2×C4).103D28, C28⋊7D4.7C2, C28.55D4⋊1C2, C14.1(C23⋊C4), C22.7(D4⋊D7), (C2×C14).37SD16, (C22×C4).52D14, C2.C42⋊6D7, C7⋊1(C22.SD16), C22.7(Q8⋊D7), C14.1(D4⋊C4), C2.3(C14.D8), C2.4(Dic14⋊C4), (C22×C14).174D4, C23.70(C7⋊D4), C22.54(D14⋊C4), (C22×C28).89C22, C2.4(C23.1D14), (C2×C4).8(C4×D7), (C2×C28).20(C2×C4), (C2×C14).33(C22⋊C4), (C7×C2.C42)⋊13C2, SmallGroup(448,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.C4≀C2
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, cac=a-1, ad=da, cbc=b-1, dbd-1=a7b, dcd-1=a7b-1c >
Subgroups: 540 in 90 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C22⋊C8, C4⋊D4, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22.SD16, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C2×D28, C2×C7⋊D4, C22×C28, C22×C28, C28.55D4, C7×C2.C42, C28⋊7D4, C14.C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, C23⋊C4, D4⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, C22.SD16, D14⋊C4, D4⋊D7, Q8⋊D7, Dic14⋊C4, C23.1D14, C14.D8, C14.C4≀C2
►On 112 points
Generators in S112
(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 47 20 31)(2 48 21 32)(3 49 22 33)(4 50 23 34)(5 51 24 35)(6 52 25 36)(7 53 26 37)(8 54 27 38)(9 55 28 39)(10 56 15 40)(11 43 16 41)(12 44 17 42)(13 45 18 29)(14 46 19 30)(57 94 77 101)(58 95 78 102)(59 96 79 103)(60 97 80 104)(61 98 81 105)(62 85 82 106)(63 86 83 107)(64 87 84 108)(65 88 71 109)(66 89 72 110)(67 90 73 111)(68 91 74 112)(69 92 75 99)(70 93 76 100)
(1 106)(2 105)(3 104)(4 103)(5 102)(6 101)(7 100)(8 99)(9 112)(10 111)(11 110)(12 109)(13 108)(14 107)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 72)(44 71)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)
(1 27)(2 28)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 43)(42 44)(57 94 84 108)(58 95 71 109)(59 96 72 110)(60 97 73 111)(61 98 74 112)(62 85 75 99)(63 86 76 100)(64 87 77 101)(65 88 78 102)(66 89 79 103)(67 90 80 104)(68 91 81 105)(69 92 82 106)(70 93 83 107)
G:=sub<Sym(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), (1,47,20,31)(2,48,21,32)(3,49,22,33)(4,50,23,34)(5,51,24,35)(6,52,25,36)(7,53,26,37)(8,54,27,38)(9,55,28,39)(10,56,15,40)(11,43,16,41)(12,44,17,42)(13,45,18,29)(14,46,19,30)(57,94,77,101)(58,95,78,102)(59,96,79,103)(60,97,80,104)(61,98,81,105)(62,85,82,106)(63,86,83,107)(64,87,84,108)(65,88,71,109)(66,89,72,110)(67,90,73,111)(68,91,74,112)(69,92,75,99)(70,93,76,100), (1,106)(2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,72)(44,71)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,43)(42,44)(57,94,84,108)(58,95,71,109)(59,96,72,110)(60,97,73,111)(61,98,74,112)(62,85,75,99)(63,86,76,100)(64,87,77,101)(65,88,78,102)(66,89,79,103)(67,90,80,104)(68,91,81,105)(69,92,82,106)(70,93,83,107)>;
G:=Group( (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), (1,47,20,31)(2,48,21,32)(3,49,22,33)(4,50,23,34)(5,51,24,35)(6,52,25,36)(7,53,26,37)(8,54,27,38)(9,55,28,39)(10,56,15,40)(11,43,16,41)(12,44,17,42)(13,45,18,29)(14,46,19,30)(57,94,77,101)(58,95,78,102)(59,96,79,103)(60,97,80,104)(61,98,81,105)(62,85,82,106)(63,86,83,107)(64,87,84,108)(65,88,71,109)(66,89,72,110)(67,90,73,111)(68,91,74,112)(69,92,75,99)(70,93,76,100), (1,106)(2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,72)(44,71)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,43)(42,44)(57,94,84,108)(58,95,71,109)(59,96,72,110)(60,97,73,111)(61,98,74,112)(62,85,75,99)(63,86,76,100)(64,87,77,101)(65,88,78,102)(66,89,79,103)(67,90,80,104)(68,91,81,105)(69,92,82,106)(70,93,83,107) );
G=PermutationGroup([[(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)], [(1,47,20,31),(2,48,21,32),(3,49,22,33),(4,50,23,34),(5,51,24,35),(6,52,25,36),(7,53,26,37),(8,54,27,38),(9,55,28,39),(10,56,15,40),(11,43,16,41),(12,44,17,42),(13,45,18,29),(14,46,19,30),(57,94,77,101),(58,95,78,102),(59,96,79,103),(60,97,80,104),(61,98,81,105),(62,85,82,106),(63,86,83,107),(64,87,84,108),(65,88,71,109),(66,89,72,110),(67,90,73,111),(68,91,74,112),(69,92,75,99),(70,93,76,100)], [(1,106),(2,105),(3,104),(4,103),(5,102),(6,101),(7,100),(8,99),(9,112),(10,111),(11,110),(12,109),(13,108),(14,107),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,98),(22,97),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,72),(44,71),(45,84),(46,83),(47,82),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73)], [(1,27),(2,28),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,43),(42,44),(57,94,84,108),(58,95,71,109),(59,96,72,110),(60,97,73,111),(61,98,74,112),(62,85,75,99),(63,86,76,100),(64,87,77,101),(65,88,78,102),(66,89,79,103),(67,90,80,104),(68,91,81,105),(69,92,82,106),(70,93,83,107)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 4 | ··· | 4 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | Dic14⋊C4 | C23⋊C4 | D4⋊D7 | Q8⋊D7 | C23.1D14 |
| kernel | C14.C4≀C2 | C28.55D4 | C7×C2.C42 | C28⋊7D4 | C4⋊Dic7 | C2×D28 | C2×C28 | C22×C14 | C2.C42 | C2×C14 | C2×C14 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 | C14 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C14.C4≀C2 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 88 | 0 | 0 |
| 0 | 0 | 25 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 2 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 34 | 0 | 0 |
| 0 | 0 | 88 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 51 |
| 0 | 0 | 0 | 0 | 95 | 69 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,25,0,0,0,0,88,34,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[15,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,0,0,0,0,0,2,15],[0,15,0,0,0,0,98,0,0,0,0,0,0,0,25,88,0,0,0,0,34,88,0,0,0,0,0,0,44,95,0,0,0,0,51,69],[112,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,15,0,0,0,0,0,112] >;
C14.C4≀C2 in GAP, Magma, Sage, TeX
C_{14}.C_4\wr C_2 % in TeX
G:=Group("C14.C4wrC2"); // GroupNames label
G:=SmallGroup(448,8);
// by ID
G=gap.SmallGroup(448,8);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,1571,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^2=d^4=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a^7*b,d*c*d^-1=a^7*b^-1*c>;
// generators/relations