direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊8D14, D28⋊12C23, C28.47C24, C14.12C25, D14.6C24, C14⋊22+ (1+4), Dic7.7C24, Dic14⋊14C23, C4○D4⋊17D14, (C2×D4)⋊50D14, (C2×C28)⋊7C23, (C2×Q8)⋊39D14, (C4×D7)⋊2C23, D4⋊9(C22×D7), C7⋊D4⋊5C23, (C7×Q8)⋊9C23, Q8⋊8(C22×D7), (D4×D7)⋊12C22, (C7×D4)⋊10C23, (C22×C4)⋊33D14, (C2×C14).3C24, C4.62(C23×D7), C2.13(D7×C24), C7⋊2(C2×2+ (1+4)), C4○D28⋊25C22, (C2×D28)⋊64C22, (D4×C14)⋊53C22, (C22×D28)⋊24C2, (Q8×C14)⋊46C22, (C22×D7)⋊5C23, (C22×C28)⋊28C22, Q8⋊2D7⋊13C22, (C23×D7)⋊18C22, C22.56(C23×D7), (C2×Dic14)⋊78C22, C23.214(C22×D7), (C22×C14).248C23, (C2×Dic7).299C23, (C2×D4×D7)⋊28C2, (C2×C4○D4)⋊13D7, (C2×C4×D7)⋊34C22, (C2×C4)⋊6(C22×D7), (C14×C4○D4)⋊14C2, (C2×C4○D28)⋊37C2, (C2×Q8⋊2D7)⋊21C2, (C7×C4○D4)⋊20C22, (C2×C7⋊D4)⋊54C22, SmallGroup(448,1376)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 4180 in 898 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×8], C4 [×4], C22, C22 [×6], C22 [×54], C7, C2×C4, C2×C4 [×15], C2×C4 [×26], D4 [×12], D4 [×60], Q8 [×4], Q8 [×4], C23 [×3], C23 [×42], D7 [×12], C14, C14 [×2], C14 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×D4 [×87], C2×Q8, C2×Q8, C4○D4 [×8], C4○D4 [×40], C24 [×6], Dic7 [×4], C28 [×8], D14 [×12], D14 [×36], C2×C14, C2×C14 [×6], C2×C14 [×6], C22×D4 [×9], C2×C4○D4, C2×C4○D4 [×5], 2+ (1+4) [×16], Dic14 [×4], C4×D7 [×24], D28 [×36], C2×Dic7 [×2], C7⋊D4 [×24], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×D7 [×30], C22×D7 [×12], C22×C14 [×3], C2×2+ (1+4), C2×Dic14, C2×C4×D7 [×6], C2×D28 [×33], C4○D28 [×24], D4×D7 [×48], Q8⋊2D7 [×16], C2×C7⋊D4 [×6], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C23×D7 [×6], C22×D28 [×3], C2×C4○D28 [×3], C2×D4×D7 [×6], C2×Q8⋊2D7 [×2], D4⋊8D14 [×16], C14×C4○D4, C2×D4⋊8D14
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⋊8D14 [×2], D7×C24, C2×D4⋊8D14
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d14=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ce=ec, ede=d-1 >
►On 112 points
Generators in S112
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 43)(8 32)(9 33)(10 34)(11 35)(12 29)(13 30)(14 31)(15 42)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 99)(69 100)(70 101)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(1 23 9 16)(2 24 10 17)(3 25 11 18)(4 26 12 19)(5 27 13 20)(6 28 14 21)(7 22 8 15)(29 39 47 52)(30 40 48 53)(31 41 49 54)(32 42 43 55)(33 36 44 56)(34 37 45 50)(35 38 46 51)(57 72 64 79)(58 73 65 80)(59 74 66 81)(60 75 67 82)(61 76 68 83)(62 77 69 84)(63 78 70 71)(85 101 92 108)(86 102 93 109)(87 103 94 110)(88 104 95 111)(89 105 96 112)(90 106 97 99)(91 107 98 100)
(1 63)(2 57)(3 65)(4 59)(5 67)(6 61)(7 69)(8 62)(9 70)(10 64)(11 58)(12 66)(13 60)(14 68)(15 84)(16 78)(17 72)(18 80)(19 74)(20 82)(21 76)(22 77)(23 71)(24 79)(25 73)(26 81)(27 75)(28 83)(29 111)(30 105)(31 99)(32 107)(33 101)(34 109)(35 103)(36 85)(37 93)(38 87)(39 95)(40 89)(41 97)(42 91)(43 100)(44 108)(45 102)(46 110)(47 104)(48 112)(49 106)(50 86)(51 94)(52 88)(53 96)(54 90)(55 98)(56 92)
(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)(30 35)(31 34)(32 33)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 56)(43 44)(45 49)(46 48)(57 61)(58 60)(62 70)(63 69)(64 68)(65 67)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,43)(8,32)(9,33)(10,34)(11,35)(12,29)(13,30)(14,31)(15,42)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,99)(69,100)(70,101)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (1,23,9,16)(2,24,10,17)(3,25,11,18)(4,26,12,19)(5,27,13,20)(6,28,14,21)(7,22,8,15)(29,39,47,52)(30,40,48,53)(31,41,49,54)(32,42,43,55)(33,36,44,56)(34,37,45,50)(35,38,46,51)(57,72,64,79)(58,73,65,80)(59,74,66,81)(60,75,67,82)(61,76,68,83)(62,77,69,84)(63,78,70,71)(85,101,92,108)(86,102,93,109)(87,103,94,110)(88,104,95,111)(89,105,96,112)(90,106,97,99)(91,107,98,100), (1,63)(2,57)(3,65)(4,59)(5,67)(6,61)(7,69)(8,62)(9,70)(10,64)(11,58)(12,66)(13,60)(14,68)(15,84)(16,78)(17,72)(18,80)(19,74)(20,82)(21,76)(22,77)(23,71)(24,79)(25,73)(26,81)(27,75)(28,83)(29,111)(30,105)(31,99)(32,107)(33,101)(34,109)(35,103)(36,85)(37,93)(38,87)(39,95)(40,89)(41,97)(42,91)(43,100)(44,108)(45,102)(46,110)(47,104)(48,112)(49,106)(50,86)(51,94)(52,88)(53,96)(54,90)(55,98)(56,92), (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)(30,35)(31,34)(32,33)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,56)(43,44)(45,49)(46,48)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,43)(8,32)(9,33)(10,34)(11,35)(12,29)(13,30)(14,31)(15,42)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,99)(69,100)(70,101)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (1,23,9,16)(2,24,10,17)(3,25,11,18)(4,26,12,19)(5,27,13,20)(6,28,14,21)(7,22,8,15)(29,39,47,52)(30,40,48,53)(31,41,49,54)(32,42,43,55)(33,36,44,56)(34,37,45,50)(35,38,46,51)(57,72,64,79)(58,73,65,80)(59,74,66,81)(60,75,67,82)(61,76,68,83)(62,77,69,84)(63,78,70,71)(85,101,92,108)(86,102,93,109)(87,103,94,110)(88,104,95,111)(89,105,96,112)(90,106,97,99)(91,107,98,100), (1,63)(2,57)(3,65)(4,59)(5,67)(6,61)(7,69)(8,62)(9,70)(10,64)(11,58)(12,66)(13,60)(14,68)(15,84)(16,78)(17,72)(18,80)(19,74)(20,82)(21,76)(22,77)(23,71)(24,79)(25,73)(26,81)(27,75)(28,83)(29,111)(30,105)(31,99)(32,107)(33,101)(34,109)(35,103)(36,85)(37,93)(38,87)(39,95)(40,89)(41,97)(42,91)(43,100)(44,108)(45,102)(46,110)(47,104)(48,112)(49,106)(50,86)(51,94)(52,88)(53,96)(54,90)(55,98)(56,92), (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)(30,35)(31,34)(32,33)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,56)(43,44)(45,49)(46,48)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,43),(8,32),(9,33),(10,34),(11,35),(12,29),(13,30),(14,31),(15,42),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,99),(69,100),(70,101),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(1,23,9,16),(2,24,10,17),(3,25,11,18),(4,26,12,19),(5,27,13,20),(6,28,14,21),(7,22,8,15),(29,39,47,52),(30,40,48,53),(31,41,49,54),(32,42,43,55),(33,36,44,56),(34,37,45,50),(35,38,46,51),(57,72,64,79),(58,73,65,80),(59,74,66,81),(60,75,67,82),(61,76,68,83),(62,77,69,84),(63,78,70,71),(85,101,92,108),(86,102,93,109),(87,103,94,110),(88,104,95,111),(89,105,96,112),(90,106,97,99),(91,107,98,100)], [(1,63),(2,57),(3,65),(4,59),(5,67),(6,61),(7,69),(8,62),(9,70),(10,64),(11,58),(12,66),(13,60),(14,68),(15,84),(16,78),(17,72),(18,80),(19,74),(20,82),(21,76),(22,77),(23,71),(24,79),(25,73),(26,81),(27,75),(28,83),(29,111),(30,105),(31,99),(32,107),(33,101),(34,109),(35,103),(36,85),(37,93),(38,87),(39,95),(40,89),(41,97),(42,91),(43,100),(44,108),(45,102),(46,110),(47,104),(48,112),(49,106),(50,86),(51,94),(52,88),(53,96),(54,90),(55,98),(56,92)], [(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),(30,35),(31,34),(32,33),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,56),(43,44),(45,49),(46,48),(57,61),(58,60),(62,70),(63,69),(64,68),(65,67),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 18 |
| 0 | 0 | 0 | 0 | 11 | 27 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 0 | 0 | 2 | 18 | 0 | 0 |
| 0 | 0 | 11 | 27 | 0 | 0 |
| 1 | 25 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 26 | 1 | 0 | 0 | 0 | 0 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 28 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 0 | 1 | 26 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,2,11,0,0,0,0,18,27,0,0,27,18,0,0,0,0,11,2,0,0],[1,24,0,0,0,0,25,21,0,0,0,0,0,0,26,21,0,0,0,0,8,21,0,0,0,0,0,0,3,8,0,0,0,0,21,8],[26,21,0,0,0,0,1,3,0,0,0,0,0,0,26,28,0,0,0,0,8,3,0,0,0,0,0,0,3,1,0,0,0,0,21,26] >;
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | ··· | 2U | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊8D14 |
| kernel | C2×D4⋊8D14 | C22×D28 | C2×C4○D28 | C2×D4×D7 | C2×Q8⋊2D7 | D4⋊8D14 | C14×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C14 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 3 | 9 | 9 | 3 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_8D_{14} % in TeX
G:=Group("C2xD4:8D14"); // GroupNames label
G:=SmallGroup(448,1376);
// by ID
G=gap.SmallGroup(448,1376);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,1684,102,18822]);
// 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=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀