direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.D14, D28⋊7C23, C28.28C24, Dic14⋊6C23, C7⋊C8⋊4C23, (C2×D4)⋊34D14, (C22×D4)⋊4D7, C14⋊4(C8⋊C22), D4⋊D7⋊17C22, (C2×C28).207D4, C28.249(C2×D4), C4.28(C23×D7), C4○D28⋊19C22, (C2×D28)⋊55C22, (D4×C14)⋊42C22, D4.D7⋊16C22, (C7×D4).20C23, D4.20(C22×D7), (C2×C28).537C23, C14.137(C22×D4), (C22×C4).268D14, (C22×C14).207D4, C23.92(C7⋊D4), C4.Dic7⋊32C22, (C2×Dic14)⋊65C22, (C22×C28).270C22, (D4×C2×C14)⋊3C2, C7⋊5(C2×C8⋊C22), (C2×D4⋊D7)⋊30C2, (C2×C7⋊C8)⋊20C22, C4.21(C2×C7⋊D4), (C2×C4○D28)⋊28C2, (C2×D4.D7)⋊30C2, (C2×C14).577(C2×D4), (C2×C4).92(C7⋊D4), (C2×C4.Dic7)⋊26C2, C2.10(C22×C7⋊D4), (C2×C4).235(C22×D7), C22.106(C2×C7⋊D4), SmallGroup(448,1246)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.D14
G = < a,b,c,d,e | a2=b4=c2=1, d14=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d13 >
Subgroups: 1172 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C22×C14, C2×C8⋊C22, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×C14, C2×C4.Dic7, C2×D4⋊D7, D4.D14, C2×D4.D7, C2×C4○D28, D4×C2×C14, C2×D4.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, C7⋊D4, C22×D7, C2×C8⋊C22, C2×C7⋊D4, C23×D7, D4.D14, C22×C7⋊D4, C2×D4.D14
►On 112 points
Generators in S112
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 85)(83 86)(84 87)
(1 8 15 22)(2 9 16 23)(3 10 17 24)(4 11 18 25)(5 12 19 26)(6 13 20 27)(7 14 21 28)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 106 99 92)(86 107 100 93)(87 108 101 94)(88 109 102 95)(89 110 103 96)(90 111 104 97)(91 112 105 98)
(1 22)(2 9)(3 24)(4 11)(5 26)(6 13)(7 28)(8 15)(10 17)(12 19)(14 21)(16 23)(18 25)(20 27)(29 50)(30 37)(31 52)(32 39)(33 54)(34 41)(35 56)(36 43)(38 45)(40 47)(42 49)(44 51)(46 53)(48 55)(58 72)(60 74)(62 76)(64 78)(66 80)(68 82)(70 84)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(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 81 15 67)(2 66 16 80)(3 79 17 65)(4 64 18 78)(5 77 19 63)(6 62 20 76)(7 75 21 61)(8 60 22 74)(9 73 23 59)(10 58 24 72)(11 71 25 57)(12 84 26 70)(13 69 27 83)(14 82 28 68)(29 96 43 110)(30 109 44 95)(31 94 45 108)(32 107 46 93)(33 92 47 106)(34 105 48 91)(35 90 49 104)(36 103 50 89)(37 88 51 102)(38 101 52 87)(39 86 53 100)(40 99 54 85)(41 112 55 98)(42 97 56 111)
G:=sub<Sym(112)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,106,99,92)(86,107,100,93)(87,108,101,94)(88,109,102,95)(89,110,103,96)(90,111,104,97)(91,112,105,98), (1,22)(2,9)(3,24)(4,11)(5,26)(6,13)(7,28)(8,15)(10,17)(12,19)(14,21)(16,23)(18,25)(20,27)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55)(58,72)(60,74)(62,76)(64,78)(66,80)(68,82)(70,84)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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), (1,81,15,67)(2,66,16,80)(3,79,17,65)(4,64,18,78)(5,77,19,63)(6,62,20,76)(7,75,21,61)(8,60,22,74)(9,73,23,59)(10,58,24,72)(11,71,25,57)(12,84,26,70)(13,69,27,83)(14,82,28,68)(29,96,43,110)(30,109,44,95)(31,94,45,108)(32,107,46,93)(33,92,47,106)(34,105,48,91)(35,90,49,104)(36,103,50,89)(37,88,51,102)(38,101,52,87)(39,86,53,100)(40,99,54,85)(41,112,55,98)(42,97,56,111)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,106,99,92)(86,107,100,93)(87,108,101,94)(88,109,102,95)(89,110,103,96)(90,111,104,97)(91,112,105,98), (1,22)(2,9)(3,24)(4,11)(5,26)(6,13)(7,28)(8,15)(10,17)(12,19)(14,21)(16,23)(18,25)(20,27)(29,50)(30,37)(31,52)(32,39)(33,54)(34,41)(35,56)(36,43)(38,45)(40,47)(42,49)(44,51)(46,53)(48,55)(58,72)(60,74)(62,76)(64,78)(66,80)(68,82)(70,84)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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), (1,81,15,67)(2,66,16,80)(3,79,17,65)(4,64,18,78)(5,77,19,63)(6,62,20,76)(7,75,21,61)(8,60,22,74)(9,73,23,59)(10,58,24,72)(11,71,25,57)(12,84,26,70)(13,69,27,83)(14,82,28,68)(29,96,43,110)(30,109,44,95)(31,94,45,108)(32,107,46,93)(33,92,47,106)(34,105,48,91)(35,90,49,104)(36,103,50,89)(37,88,51,102)(38,101,52,87)(39,86,53,100)(40,99,54,85)(41,112,55,98)(42,97,56,111) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,85),(83,86),(84,87)], [(1,8,15,22),(2,9,16,23),(3,10,17,24),(4,11,18,25),(5,12,19,26),(6,13,20,27),(7,14,21,28),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56),(57,78,71,64),(58,79,72,65),(59,80,73,66),(60,81,74,67),(61,82,75,68),(62,83,76,69),(63,84,77,70),(85,106,99,92),(86,107,100,93),(87,108,101,94),(88,109,102,95),(89,110,103,96),(90,111,104,97),(91,112,105,98)], [(1,22),(2,9),(3,24),(4,11),(5,26),(6,13),(7,28),(8,15),(10,17),(12,19),(14,21),(16,23),(18,25),(20,27),(29,50),(30,37),(31,52),(32,39),(33,54),(34,41),(35,56),(36,43),(38,45),(40,47),(42,49),(44,51),(46,53),(48,55),(58,72),(60,74),(62,76),(64,78),(66,80),(68,82),(70,84),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111)], [(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)], [(1,81,15,67),(2,66,16,80),(3,79,17,65),(4,64,18,78),(5,77,19,63),(6,62,20,76),(7,75,21,61),(8,60,22,74),(9,73,23,59),(10,58,24,72),(11,71,25,57),(12,84,26,70),(13,69,27,83),(14,82,28,68),(29,96,43,110),(30,109,44,95),(31,94,45,108),(32,107,46,93),(33,92,47,106),(34,105,48,91),(35,90,49,104),(36,103,50,89),(37,88,51,102),(38,101,52,87),(39,86,53,100),(40,99,54,85),(41,112,55,98),(42,97,56,111)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4.D14 |
| kernel | C2×D4.D14 | C2×C4.Dic7 | C2×D4⋊D7 | D4.D14 | C2×D4.D7 | C2×C4○D28 | D4×C2×C14 | C2×C28 | C22×C14 | C22×D4 | C22×C4 | C2×D4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 1 | 3 | 3 | 18 | 18 | 6 | 2 | 12 |
Matrix representation of C2×D4.D14 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 16 | 0 | 0 |
| 0 | 0 | 51 | 55 | 0 | 0 |
| 0 | 0 | 54 | 71 | 1 | 94 |
| 0 | 0 | 49 | 112 | 12 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 16 | 0 | 0 |
| 0 | 0 | 37 | 55 | 0 | 0 |
| 0 | 0 | 57 | 71 | 112 | 0 |
| 0 | 0 | 85 | 112 | 101 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 89 | 83 | 0 | 0 |
| 0 | 0 | 88 | 24 | 0 | 0 |
| 0 | 0 | 60 | 107 | 106 | 20 |
| 0 | 0 | 71 | 16 | 29 | 7 |
| 112 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 0 | 6 | 112 |
| 0 | 0 | 58 | 0 | 21 | 46 |
| 0 | 0 | 106 | 35 | 69 | 45 |
| 0 | 0 | 74 | 97 | 42 | 106 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,58,51,54,49,0,0,16,55,71,112,0,0,0,0,1,12,0,0,0,0,94,112],[1,1,0,0,0,0,0,112,0,0,0,0,0,0,58,37,57,85,0,0,16,55,71,112,0,0,0,0,112,101,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,89,88,60,71,0,0,83,24,107,16,0,0,0,0,106,29,0,0,0,0,20,7],[112,0,0,0,0,0,2,1,0,0,0,0,0,0,51,58,106,74,0,0,0,0,35,97,0,0,6,21,69,42,0,0,112,46,45,106] >;
C2×D4.D14 in GAP, Magma, Sage, TeX
C_2\times D_4.D_{14} % in TeX
G:=Group("C2xD4.D14"); // GroupNames label
G:=SmallGroup(448,1246);
// by ID
G=gap.SmallGroup(448,1246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^13>;
// generators/relations