direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×2+ (1+4), C28.90C24, C14.24C25, C24⋊6(C2×C14), (C2×C28)⋊11C23, (C7×D4)⋊15C23, D4⋊4(C22×C14), C2.4(C24×C14), (C7×Q8)⋊14C23, Q8⋊4(C22×C14), (C22×D4)⋊12C14, (D4×C14)⋊68C22, C4.13(C23×C14), (C23×C14)⋊6C22, (C22×C14)⋊4C23, C23⋊3(C22×C14), (Q8×C14)⋊60C22, (C2×C14).387C24, (C22×C28)⋊53C22, C22.2(C23×C14), (D4×C2×C14)⋊27C2, C4○D4⋊6(C2×C14), (C2×C4○D4)⋊13C14, (C14×C4○D4)⋊29C2, (C2×D4)⋊17(C2×C14), (C2×C4)⋊2(C22×C14), (C2×Q8)⋊20(C2×C14), (C22×C4)⋊13(C2×C14), (C7×C4○D4)⋊26C22, SmallGroup(448,1389)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×12], C22, C22 [×18], C22 [×42], C7, C2×C4 [×42], D4 [×72], Q8 [×8], C23 [×33], C23 [×12], C14, C14 [×2], C14 [×18], C22×C4 [×9], C2×D4 [×90], C2×Q8 [×2], C4○D4 [×48], C24 [×6], C28 [×12], C2×C14, C2×C14 [×18], C2×C14 [×42], C22×D4 [×9], C2×C4○D4 [×6], 2+ (1+4) [×16], C2×C28 [×42], C7×D4 [×72], C7×Q8 [×8], C22×C14 [×33], C22×C14 [×12], C2×2+ (1+4), C22×C28 [×9], D4×C14 [×90], Q8×C14 [×2], C7×C4○D4 [×48], C23×C14 [×6], D4×C2×C14 [×9], C14×C4○D4 [×6], C7×2+ (1+4) [×16], C14×2+ (1+4)
Quotients:
C1, C2 [×31], C22 [×155], C7, C23 [×155], C14 [×31], C24 [×31], C2×C14 [×155], 2+ (1+4) [×2], C25, C22×C14 [×155], C2×2+ (1+4), C23×C14 [×31], C7×2+ (1+4) [×2], C24×C14, C14×2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►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)
(1 106 67 16)(2 107 68 17)(3 108 69 18)(4 109 70 19)(5 110 57 20)(6 111 58 21)(7 112 59 22)(8 99 60 23)(9 100 61 24)(10 101 62 25)(11 102 63 26)(12 103 64 27)(13 104 65 28)(14 105 66 15)(29 88 78 47)(30 89 79 48)(31 90 80 49)(32 91 81 50)(33 92 82 51)(34 93 83 52)(35 94 84 53)(36 95 71 54)(37 96 72 55)(38 97 73 56)(39 98 74 43)(40 85 75 44)(41 86 76 45)(42 87 77 46)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 71)(13 72)(14 73)(15 56)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)
(1 99 67 23)(2 100 68 24)(3 101 69 25)(4 102 70 26)(5 103 57 27)(6 104 58 28)(7 105 59 15)(8 106 60 16)(9 107 61 17)(10 108 62 18)(11 109 63 19)(12 110 64 20)(13 111 65 21)(14 112 66 22)(29 54 78 95)(30 55 79 96)(31 56 80 97)(32 43 81 98)(33 44 82 85)(34 45 83 86)(35 46 84 87)(36 47 71 88)(37 48 72 89)(38 49 73 90)(39 50 74 91)(40 51 75 92)(41 52 76 93)(42 53 77 94)
(1 98)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 38)(16 39)(17 40)(18 41)(19 42)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(43 67)(44 68)(45 69)(46 70)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 99)(82 100)(83 101)(84 102)
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), (1,106,67,16)(2,107,68,17)(3,108,69,18)(4,109,70,19)(5,110,57,20)(6,111,58,21)(7,112,59,22)(8,99,60,23)(9,100,61,24)(10,101,62,25)(11,102,63,26)(12,103,64,27)(13,104,65,28)(14,105,66,15)(29,88,78,47)(30,89,79,48)(31,90,80,49)(32,91,81,50)(33,92,82,51)(34,93,83,52)(35,94,84,53)(36,95,71,54)(37,96,72,55)(38,97,73,56)(39,98,74,43)(40,85,75,44)(41,86,76,45)(42,87,77,46), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,71)(13,72)(14,73)(15,56)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,99,67,23)(2,100,68,24)(3,101,69,25)(4,102,70,26)(5,103,57,27)(6,104,58,28)(7,105,59,15)(8,106,60,16)(9,107,61,17)(10,108,62,18)(11,109,63,19)(12,110,64,20)(13,111,65,21)(14,112,66,22)(29,54,78,95)(30,55,79,96)(31,56,80,97)(32,43,81,98)(33,44,82,85)(34,45,83,86)(35,46,84,87)(36,47,71,88)(37,48,72,89)(38,49,73,90)(39,50,74,91)(40,51,75,92)(41,52,76,93)(42,53,77,94), (1,98)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)>;
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), (1,106,67,16)(2,107,68,17)(3,108,69,18)(4,109,70,19)(5,110,57,20)(6,111,58,21)(7,112,59,22)(8,99,60,23)(9,100,61,24)(10,101,62,25)(11,102,63,26)(12,103,64,27)(13,104,65,28)(14,105,66,15)(29,88,78,47)(30,89,79,48)(31,90,80,49)(32,91,81,50)(33,92,82,51)(34,93,83,52)(35,94,84,53)(36,95,71,54)(37,96,72,55)(38,97,73,56)(39,98,74,43)(40,85,75,44)(41,86,76,45)(42,87,77,46), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,71)(13,72)(14,73)(15,56)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106), (1,99,67,23)(2,100,68,24)(3,101,69,25)(4,102,70,26)(5,103,57,27)(6,104,58,28)(7,105,59,15)(8,106,60,16)(9,107,61,17)(10,108,62,18)(11,109,63,19)(12,110,64,20)(13,111,65,21)(14,112,66,22)(29,54,78,95)(30,55,79,96)(31,56,80,97)(32,43,81,98)(33,44,82,85)(34,45,83,86)(35,46,84,87)(36,47,71,88)(37,48,72,89)(38,49,73,90)(39,50,74,91)(40,51,75,92)(41,52,76,93)(42,53,77,94), (1,98)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102) );
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)], [(1,106,67,16),(2,107,68,17),(3,108,69,18),(4,109,70,19),(5,110,57,20),(6,111,58,21),(7,112,59,22),(8,99,60,23),(9,100,61,24),(10,101,62,25),(11,102,63,26),(12,103,64,27),(13,104,65,28),(14,105,66,15),(29,88,78,47),(30,89,79,48),(31,90,80,49),(32,91,81,50),(33,92,82,51),(34,93,83,52),(35,94,84,53),(36,95,71,54),(37,96,72,55),(38,97,73,56),(39,98,74,43),(40,85,75,44),(41,86,76,45),(42,87,77,46)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,71),(13,72),(14,73),(15,56),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,99),(92,100),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106)], [(1,99,67,23),(2,100,68,24),(3,101,69,25),(4,102,70,26),(5,103,57,27),(6,104,58,28),(7,105,59,15),(8,106,60,16),(9,107,61,17),(10,108,62,18),(11,109,63,19),(12,110,64,20),(13,111,65,21),(14,112,66,22),(29,54,78,95),(30,55,79,96),(31,56,80,97),(32,43,81,98),(33,44,82,85),(34,45,83,86),(35,46,84,87),(36,47,71,88),(37,48,72,89),(38,49,73,90),(39,50,74,91),(40,51,75,92),(41,52,76,93),(42,53,77,94)], [(1,98),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,38),(16,39),(17,40),(18,41),(19,42),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(43,67),(44,68),(45,69),(46,70),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,99),(82,100),(83,101),(84,102)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 2 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 28 | 0 | 28 |
| 0 | 28 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 28 |
| 0 | 0 | 28 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 28 | 1 |
| 0 | 28 | 1 | 0 | 28 |
| 0 | 28 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 28 | 28 | 0 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,1,1,0,28,0,0,0,28,0,0,0,1,0,0,0,2,1,28,28],[1,0,0,0,0,0,28,0,0,0,0,2,1,28,28,0,0,0,0,28,0,0,0,28,0],[1,0,0,0,0,0,1,0,28,28,0,0,0,1,0,0,0,28,0,0,0,2,1,28,28],[28,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,2,1,28,28,0,0,28,0,0] >;
238 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2U | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14DV | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
238 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | 2+ (1+4) | C7×2+ (1+4) |
| kernel | C14×2+ (1+4) | D4×C2×C14 | C14×C4○D4 | C7×2+ (1+4) | C2×2+ (1+4) | C22×D4 | C2×C4○D4 | 2+ (1+4) | C14 | C2 |
| # reps | 1 | 9 | 6 | 16 | 6 | 54 | 36 | 96 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}\times 2_+^{(1+4)} % in TeX
G:=Group("C14xES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1389);
// by ID
G=gap.SmallGroup(448,1389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165,2403,6499]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀