direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊6D14, C24⋊8D14, D28⋊9C23, C14.7C25, C28.42C24, D14.3C24, Dic14⋊9C23, C14⋊12+ 1+4, Dic7.3C24, (C2×D4)⋊46D14, (C2×C28)⋊5C23, D4⋊7(C22×D7), (C4×D7)⋊1C23, (C7×D4)⋊8C23, C7⋊D4⋊3C23, C2.8(D7×C24), (D4×D7)⋊11C22, (C22×D4)⋊13D7, (C22×C4)⋊32D14, C4.42(C23×D7), C23⋊4(C22×D7), C7⋊1(C2×2+ 1+4), C4○D28⋊22C22, (C2×D28)⋊61C22, (D4×C14)⋊51C22, (C22×C14)⋊7C23, (C2×Dic7)⋊5C23, (C22×D7)⋊4C23, D4⋊2D7⋊12C22, C22.8(C23×D7), (C2×C14).327C24, (C22×C28)⋊26C22, (C23×C14)⋊16C22, (C23×D7)⋊17C22, (C2×Dic14)⋊72C22, (C22×Dic7)⋊38C22, (C2×D4×D7)⋊27C2, (D4×C2×C14)⋊11C2, (C2×C4×D7)⋊33C22, (C2×C4)⋊5(C22×D7), (C2×C4○D28)⋊34C2, (C2×D4⋊2D7)⋊29C2, (C2×C7⋊D4)⋊52C22, (C22×C7⋊D4)⋊21C2, SmallGroup(448,1371)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊6D14
G = < a,b,c,d,e | a2=b4=c2=d14=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 3796 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×2+ 1+4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, D4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23×C14, C2×C4○D28, C2×D4×D7, C2×D4⋊2D7, D4⋊6D14, C22×C7⋊D4, D4×C2×C14, C2×D4⋊6D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C25, C22×D7, C2×2+ 1+4, C23×D7, D4⋊6D14, D7×C24, C2×D4⋊6D14
►On 112 points
Generators in S112
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 31)(9 32)(10 33)(11 34)(12 35)(13 29)(14 30)(15 48)(16 49)(17 43)(18 44)(19 45)(20 46)(21 47)(22 56)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(57 96)(58 97)(59 98)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)
(1 68 9 61)(2 62 10 69)(3 70 11 63)(4 64 12 57)(5 58 13 65)(6 66 14 59)(7 60 8 67)(15 105 22 112)(16 99 23 106)(17 107 24 100)(18 101 25 108)(19 109 26 102)(20 103 27 110)(21 111 28 104)(29 90 40 97)(30 98 41 91)(31 92 42 85)(32 86 36 93)(33 94 37 87)(34 88 38 95)(35 96 39 89)(43 73 51 80)(44 81 52 74)(45 75 53 82)(46 83 54 76)(47 77 55 84)(48 71 56 78)(49 79 50 72)
(1 106)(2 100)(3 108)(4 102)(5 110)(6 104)(7 112)(8 105)(9 99)(10 107)(11 101)(12 109)(13 103)(14 111)(15 67)(16 61)(17 69)(18 63)(19 57)(20 65)(21 59)(22 60)(23 68)(24 62)(25 70)(26 64)(27 58)(28 66)(29 83)(30 77)(31 71)(32 79)(33 73)(34 81)(35 75)(36 72)(37 80)(38 74)(39 82)(40 76)(41 84)(42 78)(43 94)(44 88)(45 96)(46 90)(47 98)(48 92)(49 86)(50 93)(51 87)(52 95)(53 89)(54 97)(55 91)(56 85)
(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 7)(2 6)(3 5)(8 9)(10 14)(11 13)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 34)(30 33)(31 32)(36 42)(37 41)(38 40)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(49 56)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)
G:=sub<Sym(112)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,31)(9,32)(10,33)(11,34)(12,35)(13,29)(14,30)(15,48)(16,49)(17,43)(18,44)(19,45)(20,46)(21,47)(22,56)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,96)(58,97)(59,98)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104), (1,68,9,61)(2,62,10,69)(3,70,11,63)(4,64,12,57)(5,58,13,65)(6,66,14,59)(7,60,8,67)(15,105,22,112)(16,99,23,106)(17,107,24,100)(18,101,25,108)(19,109,26,102)(20,103,27,110)(21,111,28,104)(29,90,40,97)(30,98,41,91)(31,92,42,85)(32,86,36,93)(33,94,37,87)(34,88,38,95)(35,96,39,89)(43,73,51,80)(44,81,52,74)(45,75,53,82)(46,83,54,76)(47,77,55,84)(48,71,56,78)(49,79,50,72), (1,106)(2,100)(3,108)(4,102)(5,110)(6,104)(7,112)(8,105)(9,99)(10,107)(11,101)(12,109)(13,103)(14,111)(15,67)(16,61)(17,69)(18,63)(19,57)(20,65)(21,59)(22,60)(23,68)(24,62)(25,70)(26,64)(27,58)(28,66)(29,83)(30,77)(31,71)(32,79)(33,73)(34,81)(35,75)(36,72)(37,80)(38,74)(39,82)(40,76)(41,84)(42,78)(43,94)(44,88)(45,96)(46,90)(47,98)(48,92)(49,86)(50,93)(51,87)(52,95)(53,89)(54,97)(55,91)(56,85), (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), (1,7)(2,6)(3,5)(8,9)(10,14)(11,13)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,34)(30,33)(31,32)(36,42)(37,41)(38,40)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(49,56)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,31)(9,32)(10,33)(11,34)(12,35)(13,29)(14,30)(15,48)(16,49)(17,43)(18,44)(19,45)(20,46)(21,47)(22,56)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,96)(58,97)(59,98)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104), (1,68,9,61)(2,62,10,69)(3,70,11,63)(4,64,12,57)(5,58,13,65)(6,66,14,59)(7,60,8,67)(15,105,22,112)(16,99,23,106)(17,107,24,100)(18,101,25,108)(19,109,26,102)(20,103,27,110)(21,111,28,104)(29,90,40,97)(30,98,41,91)(31,92,42,85)(32,86,36,93)(33,94,37,87)(34,88,38,95)(35,96,39,89)(43,73,51,80)(44,81,52,74)(45,75,53,82)(46,83,54,76)(47,77,55,84)(48,71,56,78)(49,79,50,72), (1,106)(2,100)(3,108)(4,102)(5,110)(6,104)(7,112)(8,105)(9,99)(10,107)(11,101)(12,109)(13,103)(14,111)(15,67)(16,61)(17,69)(18,63)(19,57)(20,65)(21,59)(22,60)(23,68)(24,62)(25,70)(26,64)(27,58)(28,66)(29,83)(30,77)(31,71)(32,79)(33,73)(34,81)(35,75)(36,72)(37,80)(38,74)(39,82)(40,76)(41,84)(42,78)(43,94)(44,88)(45,96)(46,90)(47,98)(48,92)(49,86)(50,93)(51,87)(52,95)(53,89)(54,97)(55,91)(56,85), (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), (1,7)(2,6)(3,5)(8,9)(10,14)(11,13)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,34)(30,33)(31,32)(36,42)(37,41)(38,40)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(49,56)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,31),(9,32),(10,33),(11,34),(12,35),(13,29),(14,30),(15,48),(16,49),(17,43),(18,44),(19,45),(20,46),(21,47),(22,56),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(57,96),(58,97),(59,98),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104)], [(1,68,9,61),(2,62,10,69),(3,70,11,63),(4,64,12,57),(5,58,13,65),(6,66,14,59),(7,60,8,67),(15,105,22,112),(16,99,23,106),(17,107,24,100),(18,101,25,108),(19,109,26,102),(20,103,27,110),(21,111,28,104),(29,90,40,97),(30,98,41,91),(31,92,42,85),(32,86,36,93),(33,94,37,87),(34,88,38,95),(35,96,39,89),(43,73,51,80),(44,81,52,74),(45,75,53,82),(46,83,54,76),(47,77,55,84),(48,71,56,78),(49,79,50,72)], [(1,106),(2,100),(3,108),(4,102),(5,110),(6,104),(7,112),(8,105),(9,99),(10,107),(11,101),(12,109),(13,103),(14,111),(15,67),(16,61),(17,69),(18,63),(19,57),(20,65),(21,59),(22,60),(23,68),(24,62),(25,70),(26,64),(27,58),(28,66),(29,83),(30,77),(31,71),(32,79),(33,73),(34,81),(35,75),(36,72),(37,80),(38,74),(39,82),(40,76),(41,84),(42,78),(43,94),(44,88),(45,96),(46,90),(47,98),(48,92),(49,86),(50,93),(51,87),(52,95),(53,89),(54,97),(55,91),(56,85)], [(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)], [(1,7),(2,6),(3,5),(8,9),(10,14),(11,13),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,34),(30,33),(31,32),(36,42),(37,41),(38,40),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(49,56),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,93),(86,92),(87,91),(88,90),(94,98),(95,97),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2U | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 |
| kernel | C2×D4⋊6D14 | C2×C4○D28 | C2×D4×D7 | C2×D4⋊2D7 | D4⋊6D14 | C22×C7⋊D4 | D4×C2×C14 | C22×D4 | C22×C4 | C2×D4 | C24 | C14 | C2 |
| # reps | 1 | 2 | 4 | 4 | 16 | 4 | 1 | 3 | 3 | 36 | 6 | 2 | 12 |
Matrix representation of C2×D4⋊6D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 21 | 12 |
| 0 | 0 | 26 | 22 | 18 | 12 |
| 0 | 0 | 9 | 20 | 24 | 13 |
| 0 | 0 | 1 | 23 | 16 | 5 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 14 |
| 0 | 0 | 0 | 28 | 15 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 25 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 8 | 0 | 0 |
| 0 | 0 | 13 | 20 | 0 | 0 |
| 0 | 0 | 17 | 17 | 3 | 21 |
| 0 | 0 | 8 | 26 | 8 | 8 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 28 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 25 | 18 | 0 | 0 |
| 0 | 0 | 17 | 17 | 3 | 21 |
| 0 | 0 | 14 | 25 | 1 | 26 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,7,26,9,1,0,0,16,22,20,23,0,0,21,18,24,16,0,0,12,12,13,5],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,15,1,0,0,0,14,4,0,1],[18,4,0,0,0,0,25,4,0,0,0,0,0,0,27,13,17,8,0,0,8,20,17,26,0,0,0,0,3,8,0,0,0,0,21,8],[11,28,0,0,0,0,4,18,0,0,0,0,0,0,11,25,17,14,0,0,1,18,17,25,0,0,0,0,3,1,0,0,0,0,21,26] >;
C2×D4⋊6D14 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_6D_{14} % in TeX
G:=Group("C2xD4:6D14"); // GroupNames label
G:=SmallGroup(448,1371);
// by ID
G=gap.SmallGroup(448,1371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations