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, (C4×D7)⋊1C23, (C7×D4)⋊8C23, D4⋊7(C22×D7), C7⋊D4⋊3C23, C2.8(D7×C24), (C22×D4)⋊13D7, (D4×D7)⋊11C22, (C22×C4)⋊32D14, C4.42(C23×D7), C23⋊4(C22×D7), C7⋊1(C2×2+ (1+4)), C4○D28⋊22C22, (D4×C14)⋊51C22, (C2×D28)⋊61C22, (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
Subgroups: 3796 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×50], C7, C2×C4 [×6], C2×C4 [×36], D4 [×16], D4 [×56], Q8 [×8], C23, C23 [×12], C23 [×32], D7 [×8], C14, C14 [×2], C14 [×10], C22×C4, C22×C4 [×8], C2×D4 [×12], C2×D4 [×78], C2×Q8 [×2], C4○D4 [×48], C24 [×2], C24 [×4], Dic7 [×8], C28 [×4], D14 [×8], D14 [×24], C2×C14, C2×C14 [×10], C2×C14 [×18], C22×D4, C22×D4 [×8], C2×C4○D4 [×6], 2+ (1+4) [×16], Dic14 [×8], C4×D7 [×16], D28 [×8], C2×Dic7 [×20], C7⋊D4 [×48], C2×C28 [×6], C7×D4 [×16], C22×D7 [×20], C22×D7 [×8], C22×C14, C22×C14 [×12], C22×C14 [×4], C2×2+ (1+4), C2×Dic14 [×2], C2×C4×D7 [×4], C2×D28 [×2], C4○D28 [×16], D4×D7 [×32], D4⋊2D7 [×32], C22×Dic7 [×4], C2×C7⋊D4 [×44], C22×C28, D4×C14 [×12], C23×D7 [×4], C23×C14 [×2], C2×C4○D28 [×2], C2×D4×D7 [×4], C2×D4⋊2D7 [×4], D4⋊6D14 [×16], C22×C7⋊D4 [×4], D4×C2×C14, C2×D4⋊6D14
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], 2+ (1+4) [×2], C25, C22×D7 [×35], C2×2+ (1+4), C23×D7 [×15], D4⋊6D14 [×2], D7×C24, C2×D4⋊6D14
Generators and relations
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 >
►On 112 points
Generators in S112
(1 10)(2 11)(3 12)(4 13)(5 14)(6 8)(7 9)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 43)(22 54)(23 55)(24 56)(25 50)(26 51)(27 52)(28 53)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 83 35 76)(2 77 29 84)(3 71 30 78)(4 79 31 72)(5 73 32 80)(6 81 33 74)(7 75 34 82)(8 60 36 67)(9 68 37 61)(10 62 38 69)(11 70 39 63)(12 64 40 57)(13 58 41 65)(14 66 42 59)(15 93 22 86)(16 87 23 94)(17 95 24 88)(18 89 25 96)(19 97 26 90)(20 91 27 98)(21 85 28 92)(43 99 53 106)(44 107 54 100)(45 101 55 108)(46 109 56 102)(47 103 50 110)(48 111 51 104)(49 105 52 112)
(1 109)(2 103)(3 111)(4 105)(5 99)(6 107)(7 101)(8 93)(9 87)(10 95)(11 89)(12 97)(13 91)(14 85)(15 60)(16 68)(17 62)(18 70)(19 64)(20 58)(21 66)(22 67)(23 61)(24 69)(25 63)(26 57)(27 65)(28 59)(29 110)(30 104)(31 112)(32 106)(33 100)(34 108)(35 102)(36 86)(37 94)(38 88)(39 96)(40 90)(41 98)(42 92)(43 73)(44 81)(45 75)(46 83)(47 77)(48 71)(49 79)(50 84)(51 78)(52 72)(53 80)(54 74)(55 82)(56 76)
(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 5)(2 4)(6 7)(8 9)(10 14)(11 13)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 31)(32 35)(33 34)(36 37)(38 42)(39 41)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 88)(86 87)(89 98)(90 97)(91 96)(92 95)(93 94)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,8)(7,9)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,54)(23,55)(24,56)(25,50)(26,51)(27,52)(28,53)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,83,35,76)(2,77,29,84)(3,71,30,78)(4,79,31,72)(5,73,32,80)(6,81,33,74)(7,75,34,82)(8,60,36,67)(9,68,37,61)(10,62,38,69)(11,70,39,63)(12,64,40,57)(13,58,41,65)(14,66,42,59)(15,93,22,86)(16,87,23,94)(17,95,24,88)(18,89,25,96)(19,97,26,90)(20,91,27,98)(21,85,28,92)(43,99,53,106)(44,107,54,100)(45,101,55,108)(46,109,56,102)(47,103,50,110)(48,111,51,104)(49,105,52,112), (1,109)(2,103)(3,111)(4,105)(5,99)(6,107)(7,101)(8,93)(9,87)(10,95)(11,89)(12,97)(13,91)(14,85)(15,60)(16,68)(17,62)(18,70)(19,64)(20,58)(21,66)(22,67)(23,61)(24,69)(25,63)(26,57)(27,65)(28,59)(29,110)(30,104)(31,112)(32,106)(33,100)(34,108)(35,102)(36,86)(37,94)(38,88)(39,96)(40,90)(41,98)(42,92)(43,73)(44,81)(45,75)(46,83)(47,77)(48,71)(49,79)(50,84)(51,78)(52,72)(53,80)(54,74)(55,82)(56,76), (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), (1,5)(2,4)(6,7)(8,9)(10,14)(11,13)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,31)(32,35)(33,34)(36,37)(38,42)(39,41)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,8)(7,9)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,54)(23,55)(24,56)(25,50)(26,51)(27,52)(28,53)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,83,35,76)(2,77,29,84)(3,71,30,78)(4,79,31,72)(5,73,32,80)(6,81,33,74)(7,75,34,82)(8,60,36,67)(9,68,37,61)(10,62,38,69)(11,70,39,63)(12,64,40,57)(13,58,41,65)(14,66,42,59)(15,93,22,86)(16,87,23,94)(17,95,24,88)(18,89,25,96)(19,97,26,90)(20,91,27,98)(21,85,28,92)(43,99,53,106)(44,107,54,100)(45,101,55,108)(46,109,56,102)(47,103,50,110)(48,111,51,104)(49,105,52,112), (1,109)(2,103)(3,111)(4,105)(5,99)(6,107)(7,101)(8,93)(9,87)(10,95)(11,89)(12,97)(13,91)(14,85)(15,60)(16,68)(17,62)(18,70)(19,64)(20,58)(21,66)(22,67)(23,61)(24,69)(25,63)(26,57)(27,65)(28,59)(29,110)(30,104)(31,112)(32,106)(33,100)(34,108)(35,102)(36,86)(37,94)(38,88)(39,96)(40,90)(41,98)(42,92)(43,73)(44,81)(45,75)(46,83)(47,77)(48,71)(49,79)(50,84)(51,78)(52,72)(53,80)(54,74)(55,82)(56,76), (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), (1,5)(2,4)(6,7)(8,9)(10,14)(11,13)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,31)(32,35)(33,34)(36,37)(38,42)(39,41)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,13),(5,14),(6,8),(7,9),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,43),(22,54),(23,55),(24,56),(25,50),(26,51),(27,52),(28,53),(29,39),(30,40),(31,41),(32,42),(33,36),(34,37),(35,38),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(1,83,35,76),(2,77,29,84),(3,71,30,78),(4,79,31,72),(5,73,32,80),(6,81,33,74),(7,75,34,82),(8,60,36,67),(9,68,37,61),(10,62,38,69),(11,70,39,63),(12,64,40,57),(13,58,41,65),(14,66,42,59),(15,93,22,86),(16,87,23,94),(17,95,24,88),(18,89,25,96),(19,97,26,90),(20,91,27,98),(21,85,28,92),(43,99,53,106),(44,107,54,100),(45,101,55,108),(46,109,56,102),(47,103,50,110),(48,111,51,104),(49,105,52,112)], [(1,109),(2,103),(3,111),(4,105),(5,99),(6,107),(7,101),(8,93),(9,87),(10,95),(11,89),(12,97),(13,91),(14,85),(15,60),(16,68),(17,62),(18,70),(19,64),(20,58),(21,66),(22,67),(23,61),(24,69),(25,63),(26,57),(27,65),(28,59),(29,110),(30,104),(31,112),(32,106),(33,100),(34,108),(35,102),(36,86),(37,94),(38,88),(39,96),(40,90),(41,98),(42,92),(43,73),(44,81),(45,75),(46,83),(47,77),(48,71),(49,79),(50,84),(51,78),(52,72),(53,80),(54,74),(55,82),(56,76)], [(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)], [(1,5),(2,4),(6,7),(8,9),(10,14),(11,13),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,31),(32,35),(33,34),(36,37),(38,42),(39,41),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,88),(86,87),(89,98),(90,97),(91,96),(92,95),(93,94),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀