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, (D4×C14)⋊42C22, (C2×D28)⋊55C22, D4.D7⋊16C22, D4.20(C22×D7), (C7×D4).20C23, (C2×C28).537C23, C14.137(C22×D4), (C22×C14).207D4, (C22×C4).268D14, 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
Subgroups: 1172 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×4], D4 [×13], Q8 [×3], C23, C23 [×11], D7 [×2], C14, C14 [×2], C14 [×6], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×5], C2×Q8, C4○D4 [×6], C24, Dic7 [×2], C28 [×2], C28 [×2], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×18], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C7⋊C8 [×4], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7, C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C7×D4 [×4], C7×D4 [×6], C22×D7, C22×C14, C22×C14 [×10], C2×C8⋊C22, C2×C7⋊C8 [×2], C4.Dic7 [×4], D4⋊D7 [×8], D4.D7 [×8], C2×Dic14, C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C2×C7⋊D4, C22×C28, D4×C14 [×6], D4×C14 [×3], C23×C14, C2×C4.Dic7, C2×D4⋊D7 [×2], D4.D14 [×8], C2×D4.D7 [×2], C2×C4○D28, D4×C2×C14, C2×D4.D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8⋊C22 [×2], C22×D4, C7⋊D4 [×4], C22×D7 [×7], C2×C8⋊C22, C2×C7⋊D4 [×6], C23×D7, D4.D14 [×2], C22×C7⋊D4, C2×D4.D14
Generators and relations
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 >
►On 112 points
Generators in S112
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 29)(27 30)(28 31)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)
(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 36)(30 51)(31 38)(32 53)(33 40)(34 55)(35 42)(37 44)(39 46)(41 48)(43 50)(45 52)(47 54)(49 56)(57 71)(59 73)(61 75)(63 77)(65 79)(67 81)(69 83)(86 100)(88 102)(90 104)(92 106)(94 108)(96 110)(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 64 15 78)(2 77 16 63)(3 62 17 76)(4 75 18 61)(5 60 19 74)(6 73 20 59)(7 58 21 72)(8 71 22 57)(9 84 23 70)(10 69 24 83)(11 82 25 68)(12 67 26 81)(13 80 27 66)(14 65 28 79)(29 98 43 112)(30 111 44 97)(31 96 45 110)(32 109 46 95)(33 94 47 108)(34 107 48 93)(35 92 49 106)(36 105 50 91)(37 90 51 104)(38 103 52 89)(39 88 53 102)(40 101 54 87)(41 86 55 100)(42 99 56 85)
G:=sub<Sym(112)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101), (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)(29,36)(30,51)(31,38)(32,53)(33,40)(34,55)(35,42)(37,44)(39,46)(41,48)(43,50)(45,52)(47,54)(49,56)(57,71)(59,73)(61,75)(63,77)(65,79)(67,81)(69,83)(86,100)(88,102)(90,104)(92,106)(94,108)(96,110)(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), (1,64,15,78)(2,77,16,63)(3,62,17,76)(4,75,18,61)(5,60,19,74)(6,73,20,59)(7,58,21,72)(8,71,22,57)(9,84,23,70)(10,69,24,83)(11,82,25,68)(12,67,26,81)(13,80,27,66)(14,65,28,79)(29,98,43,112)(30,111,44,97)(31,96,45,110)(32,109,46,95)(33,94,47,108)(34,107,48,93)(35,92,49,106)(36,105,50,91)(37,90,51,104)(38,103,52,89)(39,88,53,102)(40,101,54,87)(41,86,55,100)(42,99,56,85)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101), (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)(29,36)(30,51)(31,38)(32,53)(33,40)(34,55)(35,42)(37,44)(39,46)(41,48)(43,50)(45,52)(47,54)(49,56)(57,71)(59,73)(61,75)(63,77)(65,79)(67,81)(69,83)(86,100)(88,102)(90,104)(92,106)(94,108)(96,110)(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), (1,64,15,78)(2,77,16,63)(3,62,17,76)(4,75,18,61)(5,60,19,74)(6,73,20,59)(7,58,21,72)(8,71,22,57)(9,84,23,70)(10,69,24,83)(11,82,25,68)(12,67,26,81)(13,80,27,66)(14,65,28,79)(29,98,43,112)(30,111,44,97)(31,96,45,110)(32,109,46,95)(33,94,47,108)(34,107,48,93)(35,92,49,106)(36,105,50,91)(37,90,51,104)(38,103,52,89)(39,88,53,102)(40,101,54,87)(41,86,55,100)(42,99,56,85) );
G=PermutationGroup([(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,29),(27,30),(28,31),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101)], [(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),(29,36),(30,51),(31,38),(32,53),(33,40),(34,55),(35,42),(37,44),(39,46),(41,48),(43,50),(45,52),(47,54),(49,56),(57,71),(59,73),(61,75),(63,77),(65,79),(67,81),(69,83),(86,100),(88,102),(90,104),(92,106),(94,108),(96,110),(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)], [(1,64,15,78),(2,77,16,63),(3,62,17,76),(4,75,18,61),(5,60,19,74),(6,73,20,59),(7,58,21,72),(8,71,22,57),(9,84,23,70),(10,69,24,83),(11,82,25,68),(12,67,26,81),(13,80,27,66),(14,65,28,79),(29,98,43,112),(30,111,44,97),(31,96,45,110),(32,109,46,95),(33,94,47,108),(34,107,48,93),(35,92,49,106),(36,105,50,91),(37,90,51,104),(38,103,52,89),(39,88,53,102),(40,101,54,87),(41,86,55,100),(42,99,56,85)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀