metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊11D4, C42⋊27D14, C14.772+ (1+4), C7⋊4(D42), C4⋊2(D4×D7), C28⋊3(C2×D4), D14⋊8(C2×D4), C4⋊1D4⋊6D7, (C2×D4)⋊26D14, (C4×D28)⋊49C2, C28⋊2D4⋊36C2, (C4×C28)⋊27C22, C23⋊D14⋊27C2, D14⋊C4⋊70C22, (D4×C14)⋊33C22, C4⋊Dic7⋊74C22, C14.95(C22×D4), (C2×C28).509C23, (C2×C14).261C24, (C23×D7)⋊13C22, C23.D7⋊37C22, C2.81(D4⋊6D14), C23.67(C22×D7), (C2×D28).269C22, (C22×C14).75C23, C22.282(C23×D7), (C2×Dic7).136C23, (C22×D7).229C23, (C2×D4×D7)⋊20C2, C2.68(C2×D4×D7), (C7×C4⋊1D4)⋊8C2, (C2×C4×D7)⋊29C22, (C2×C7⋊D4)⋊27C22, (C2×C4).214(C22×D7), SmallGroup(448,1170)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2572 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×44], C7, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×4], C23 [×24], D7 [×8], C14, C14 [×2], C14 [×4], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×D4 [×26], C24 [×4], Dic7 [×4], C28 [×4], C28, D14 [×8], D14 [×24], C2×C14, C2×C14 [×12], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×D7 [×8], D28 [×8], C2×Dic7 [×4], C7⋊D4 [×16], C2×C28, C2×C28 [×2], C7×D4 [×10], C22×D7 [×4], C22×D7 [×20], C22×C14 [×4], D42, C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×4], C4×C28, C2×C4×D7 [×4], C2×D28 [×2], D4×D7 [×16], C2×C7⋊D4 [×8], D4×C14 [×6], C23×D7 [×4], C4×D28 [×2], C23⋊D14 [×4], C28⋊2D4 [×4], C7×C4⋊1D4, C2×D4×D7 [×4], D28⋊11D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D7, C2×D4 [×12], C24, D14 [×7], C22×D4 [×2], 2+ (1+4), C22×D7 [×7], D42, D4×D7 [×4], C23×D7, C2×D4×D7 [×2], D4⋊6D14, D28⋊11D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, dad=a15, cbc-1=a14b, bd=db, dcd=c-1 >
►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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(77 84)(78 83)(79 82)(80 81)(85 90)(86 89)(87 88)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)
(1 88 60 35)(2 89 61 36)(3 90 62 37)(4 91 63 38)(5 92 64 39)(6 93 65 40)(7 94 66 41)(8 95 67 42)(9 96 68 43)(10 97 69 44)(11 98 70 45)(12 99 71 46)(13 100 72 47)(14 101 73 48)(15 102 74 49)(16 103 75 50)(17 104 76 51)(18 105 77 52)(19 106 78 53)(20 107 79 54)(21 108 80 55)(22 109 81 56)(23 110 82 29)(24 111 83 30)(25 112 84 31)(26 85 57 32)(27 86 58 33)(28 87 59 34)
(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)(29 103)(30 90)(31 105)(32 92)(33 107)(34 94)(35 109)(36 96)(37 111)(38 98)(39 85)(40 100)(41 87)(42 102)(43 89)(44 104)(45 91)(46 106)(47 93)(48 108)(49 95)(50 110)(51 97)(52 112)(53 99)(54 86)(55 101)(56 88)(57 64)(58 79)(59 66)(60 81)(61 68)(62 83)(63 70)(65 72)(67 74)(69 76)(71 78)(73 80)(75 82)(77 84)
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,90)(86,89)(87,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102), (1,88,60,35)(2,89,61,36)(3,90,62,37)(4,91,63,38)(5,92,64,39)(6,93,65,40)(7,94,66,41)(8,95,67,42)(9,96,68,43)(10,97,69,44)(11,98,70,45)(12,99,71,46)(13,100,72,47)(14,101,73,48)(15,102,74,49)(16,103,75,50)(17,104,76,51)(18,105,77,52)(19,106,78,53)(20,107,79,54)(21,108,80,55)(22,109,81,56)(23,110,82,29)(24,111,83,30)(25,112,84,31)(26,85,57,32)(27,86,58,33)(28,87,59,34), (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)(29,103)(30,90)(31,105)(32,92)(33,107)(34,94)(35,109)(36,96)(37,111)(38,98)(39,85)(40,100)(41,87)(42,102)(43,89)(44,104)(45,91)(46,106)(47,93)(48,108)(49,95)(50,110)(51,97)(52,112)(53,99)(54,86)(55,101)(56,88)(57,64)(58,79)(59,66)(60,81)(61,68)(62,83)(63,70)(65,72)(67,74)(69,76)(71,78)(73,80)(75,82)(77,84)>;
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,90)(86,89)(87,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102), (1,88,60,35)(2,89,61,36)(3,90,62,37)(4,91,63,38)(5,92,64,39)(6,93,65,40)(7,94,66,41)(8,95,67,42)(9,96,68,43)(10,97,69,44)(11,98,70,45)(12,99,71,46)(13,100,72,47)(14,101,73,48)(15,102,74,49)(16,103,75,50)(17,104,76,51)(18,105,77,52)(19,106,78,53)(20,107,79,54)(21,108,80,55)(22,109,81,56)(23,110,82,29)(24,111,83,30)(25,112,84,31)(26,85,57,32)(27,86,58,33)(28,87,59,34), (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)(29,103)(30,90)(31,105)(32,92)(33,107)(34,94)(35,109)(36,96)(37,111)(38,98)(39,85)(40,100)(41,87)(42,102)(43,89)(44,104)(45,91)(46,106)(47,93)(48,108)(49,95)(50,110)(51,97)(52,112)(53,99)(54,86)(55,101)(56,88)(57,64)(58,79)(59,66)(60,81)(61,68)(62,83)(63,70)(65,72)(67,74)(69,76)(71,78)(73,80)(75,82)(77,84) );
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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(77,84),(78,83),(79,82),(80,81),(85,90),(86,89),(87,88),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102)], [(1,88,60,35),(2,89,61,36),(3,90,62,37),(4,91,63,38),(5,92,64,39),(6,93,65,40),(7,94,66,41),(8,95,67,42),(9,96,68,43),(10,97,69,44),(11,98,70,45),(12,99,71,46),(13,100,72,47),(14,101,73,48),(15,102,74,49),(16,103,75,50),(17,104,76,51),(18,105,77,52),(19,106,78,53),(20,107,79,54),(21,108,80,55),(22,109,81,56),(23,110,82,29),(24,111,83,30),(25,112,84,31),(26,85,57,32),(27,86,58,33),(28,87,59,34)], [(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),(29,103),(30,90),(31,105),(32,92),(33,107),(34,94),(35,109),(36,96),(37,111),(38,98),(39,85),(40,100),(41,87),(42,102),(43,89),(44,104),(45,91),(46,106),(47,93),(48,108),(49,95),(50,110),(51,97),(52,112),(53,99),(54,86),(55,101),(56,88),(57,64),(58,79),(59,66),(60,81),(61,68),(62,83),(63,70),(65,72),(67,74),(69,76),(71,78),(73,80),(75,82),(77,84)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 8 | 26 | 0 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 26 | 0 | 0 | 0 | 0 |
| 21 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [8,3,0,0,0,0,26,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[8,21,0,0,0,0,26,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 |
| kernel | D28⋊11D4 | C4×D28 | C23⋊D14 | C28⋊2D4 | C7×C4⋊1D4 | C2×D4×D7 | D28 | C4⋊1D4 | C42 | C2×D4 | C14 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 1 | 4 | 8 | 3 | 3 | 18 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{11}D_4 % in TeX
G:=Group("D28:11D4"); // GroupNames label
G:=SmallGroup(448,1170);
// by ID
G=gap.SmallGroup(448,1170);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,c*b*c^-1=a^14*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀