metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.35C24, D28.31C23, Dic14.30C23, C4○D4⋊16D14, (C2×D4)⋊42D14, (C2×Q8)⋊31D14, C7⋊C8.14C23, D4⋊D7⋊19C22, C28.427(C2×D4), (C2×C28).218D4, Q8⋊D7⋊18C22, D4⋊D14⋊13C2, C4.35(C23×D7), D4.8D14⋊7C2, C4○D28⋊21C22, (C2×D28)⋊59C22, (D4×C14)⋊46C22, C7⋊5(D8⋊C22), D4.D7⋊17C22, (Q8×C14)⋊38C22, (C7×D4).23C23, D4.23(C22×D7), C7⋊Q16⋊16C22, D4.D14⋊13C2, D4.9D14⋊13C2, (C7×Q8).23C23, Q8.23(C22×D7), C28.C23⋊13C2, (C2×C28).557C23, (C22×C4).282D14, C14.160(C22×D4), (C22×C14).124D4, C23.34(C7⋊D4), C4.Dic7⋊37C22, (C2×Dic14)⋊69C22, (C22×C28).292C22, (C2×C4○D4)⋊4D7, (C14×C4○D4)⋊4C2, (C2×C7⋊C8)⋊23C22, (C2×C4○D28)⋊31C2, C4.121(C2×C7⋊D4), (C2×C14).591(C2×D4), (C7×C4○D4)⋊18C22, (C2×C4.Dic7)⋊31C2, C2.33(C22×C7⋊D4), C22.21(C2×C7⋊D4), (C2×C4).203(C7⋊D4), (C2×C4).246(C22×D7), SmallGroup(448,1275)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1044 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×9], C7, C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D7 [×2], C14, C14 [×5], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic7 [×2], C28 [×4], C28 [×2], D14 [×4], C2×C14 [×3], C2×C14 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4, C2×C4○D4, C7⋊C8 [×4], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7, C7⋊D4 [×4], C2×C28 [×6], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×D7, C22×C14, C22×C14, D8⋊C22, C2×C7⋊C8 [×2], C4.Dic7 [×4], D4⋊D7 [×4], D4.D7 [×4], Q8⋊D7 [×4], C7⋊Q16 [×4], C2×Dic14, C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], C2×C4.Dic7, D4.D14 [×2], C28.C23 [×2], D4⋊D14 [×2], D4.8D14 [×4], D4.9D14 [×2], C2×C4○D28, C14×C4○D4, C28.C24
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C7⋊D4 [×4], C22×D7 [×7], D8⋊C22, C2×C7⋊D4 [×6], C23×D7, C22×C7⋊D4, C28.C24
Generators and relations
G = < a,b,c,d,e | a28=b2=c2=e2=1, d2=a14, bab=a-1, ac=ca, ad=da, eae=a15, bc=cb, bd=db, ebe=a21b, cd=dc, ece=a14c, de=ed >
►On 112 points
(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)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 30)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 86)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(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 79 15 65)(2 80 16 66)(3 81 17 67)(4 82 18 68)(5 83 19 69)(6 84 20 70)(7 57 21 71)(8 58 22 72)(9 59 23 73)(10 60 24 74)(11 61 25 75)(12 62 26 76)(13 63 27 77)(14 64 28 78)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)
(1 47)(2 34)(3 49)(4 36)(5 51)(6 38)(7 53)(8 40)(9 55)(10 42)(11 29)(12 44)(13 31)(14 46)(15 33)(16 48)(17 35)(18 50)(19 37)(20 52)(21 39)(22 54)(23 41)(24 56)(25 43)(26 30)(27 45)(28 32)(57 95)(58 110)(59 97)(60 112)(61 99)(62 86)(63 101)(64 88)(65 103)(66 90)(67 105)(68 92)(69 107)(70 94)(71 109)(72 96)(73 111)(74 98)(75 85)(76 100)(77 87)(78 102)(79 89)(80 104)(81 91)(82 106)(83 93)(84 108)
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), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,86)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(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), (1,79,15,65)(2,80,16,66)(3,81,17,67)(4,82,18,68)(5,83,19,69)(6,84,20,70)(7,57,21,71)(8,58,22,72)(9,59,23,73)(10,60,24,74)(11,61,25,75)(12,62,26,76)(13,63,27,77)(14,64,28,78)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (1,47)(2,34)(3,49)(4,36)(5,51)(6,38)(7,53)(8,40)(9,55)(10,42)(11,29)(12,44)(13,31)(14,46)(15,33)(16,48)(17,35)(18,50)(19,37)(20,52)(21,39)(22,54)(23,41)(24,56)(25,43)(26,30)(27,45)(28,32)(57,95)(58,110)(59,97)(60,112)(61,99)(62,86)(63,101)(64,88)(65,103)(66,90)(67,105)(68,92)(69,107)(70,94)(71,109)(72,96)(73,111)(74,98)(75,85)(76,100)(77,87)(78,102)(79,89)(80,104)(81,91)(82,106)(83,93)(84,108)>;
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), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,86)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(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), (1,79,15,65)(2,80,16,66)(3,81,17,67)(4,82,18,68)(5,83,19,69)(6,84,20,70)(7,57,21,71)(8,58,22,72)(9,59,23,73)(10,60,24,74)(11,61,25,75)(12,62,26,76)(13,63,27,77)(14,64,28,78)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (1,47)(2,34)(3,49)(4,36)(5,51)(6,38)(7,53)(8,40)(9,55)(10,42)(11,29)(12,44)(13,31)(14,46)(15,33)(16,48)(17,35)(18,50)(19,37)(20,52)(21,39)(22,54)(23,41)(24,56)(25,43)(26,30)(27,45)(28,32)(57,95)(58,110)(59,97)(60,112)(61,99)(62,86)(63,101)(64,88)(65,103)(66,90)(67,105)(68,92)(69,107)(70,94)(71,109)(72,96)(73,111)(74,98)(75,85)(76,100)(77,87)(78,102)(79,89)(80,104)(81,91)(82,106)(83,93)(84,108) );
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)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,30),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(74,84),(75,83),(76,82),(77,81),(78,80),(85,86),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100)], [(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(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)], [(1,79,15,65),(2,80,16,66),(3,81,17,67),(4,82,18,68),(5,83,19,69),(6,84,20,70),(7,57,21,71),(8,58,22,72),(9,59,23,73),(10,60,24,74),(11,61,25,75),(12,62,26,76),(13,63,27,77),(14,64,28,78),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98)], [(1,47),(2,34),(3,49),(4,36),(5,51),(6,38),(7,53),(8,40),(9,55),(10,42),(11,29),(12,44),(13,31),(14,46),(15,33),(16,48),(17,35),(18,50),(19,37),(20,52),(21,39),(22,54),(23,41),(24,56),(25,43),(26,30),(27,45),(28,32),(57,95),(58,110),(59,97),(60,112),(61,99),(62,86),(63,101),(64,88),(65,103),(66,90),(67,105),(68,92),(69,107),(70,94),(71,109),(72,96),(73,111),(74,98),(75,85),(76,100),(77,87),(78,102),(79,89),(80,104),(81,91),(82,106),(83,93),(84,108)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 17 | 36 | 0 | 0 |
| 77 | 90 | 0 | 0 |
| 96 | 77 | 96 | 77 |
| 36 | 23 | 36 | 23 |
| 1 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 |
| 8 | 4 | 17 | 8 |
| 83 | 105 | 77 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 112 | 0 |
| 0 | 112 | 0 | 112 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 112 | 0 | 111 | 0 |
| 0 | 112 | 0 | 111 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [17,77,96,36,36,90,77,23,0,0,96,36,0,0,77,23],[1,24,8,83,0,112,4,105,0,0,17,77,0,0,8,96],[1,0,112,0,0,1,0,112,0,0,112,0,0,0,0,112],[98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[112,0,0,0,0,112,0,0,111,0,1,0,0,111,0,1] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D8⋊C22 | C28.C24 |
| kernel | C28.C24 | C2×C4.Dic7 | D4.D14 | C28.C23 | D4⋊D14 | D4.8D14 | D4.9D14 | C2×C4○D28 | C14×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{28}.C_2^4 % in TeX
G:=Group("C28.C2^4"); // GroupNames label
G:=SmallGroup(448,1275);
// by ID
G=gap.SmallGroup(448,1275);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,570,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=c^2=e^2=1,d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^15,b*c=c*b,b*d=d*b,e*b*e=a^21*b,c*d=d*c,e*c*e=a^14*c,d*e=e*d>;
// generators/relations