metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4:C4:36D14, (C2xD28):11C4, C4.62(C2xD28), (C2xC4).46D28, D28.25(C2xC4), (C2xC28).472D4, C28.142(C2xD4), C42:C2:1D7, C14.D8:27C2, C4.23(D14:C4), C28.64(C22xC4), (C22xC14).74D4, C2.2(D4:D14), C28.47(C22:C4), (C2xC28).328C23, (C22xD28).12C2, (C22xC4).110D14, C23.54(C7:D4), C7:2(C23.37D4), C14.105(C8:C22), C22.23(D14:C4), (C2xD28).234C22, (C22xC28).150C22, C4.51(C2xC4xD7), (C2xC7:C8):4C22, (C2xC4).44(C4xD7), (C7xC4:C4):41C22, (C2xC28).91(C2xC4), C2.17(C2xD14:C4), (C2xC4.Dic7):9C2, (C7xC42:C2):1C2, (C2xC14).457(C2xD4), C14.44(C2xC22:C4), C22.72(C2xC7:D4), (C2xC4).241(C7:D4), (C2xC4).428(C22xD7), (C2xC14).14(C22:C4), SmallGroup(448,535)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:C4:36D14
G = < a,b,c,d | a4=b4=c14=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab-1, dcd=c-1 >
Subgroups: 1268 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, D7, C14, C14, C14, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C2xD4, C24, C28, C28, C28, D14, C2xC14, C2xC14, C2xC14, D4:C4, C42:C2, C2xM4(2), C22xD4, C7:C8, D28, D28, C2xC28, C2xC28, C2xC28, C22xD7, C22xC14, C23.37D4, C2xC7:C8, C4.Dic7, C4xC28, C7xC22:C4, C7xC4:C4, C2xD28, C2xD28, C22xC28, C23xD7, C14.D8, C2xC4.Dic7, C7xC42:C2, C22xD28, C4:C4:36D14
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22:C4, C22xC4, C2xD4, D14, C2xC22:C4, C8:C22, C4xD7, D28, C7:D4, C22xD7, C23.37D4, D14:C4, C2xC4xD7, C2xD28, C2xC7:D4, C2xD14:C4, D4:D14, C4:C4:36D14
►On 112 points
Generators in S112
(1 47 10 54)(2 48 11 55)(3 49 12 56)(4 43 13 50)(5 44 14 51)(6 45 8 52)(7 46 9 53)(15 36 22 29)(16 37 23 30)(17 38 24 31)(18 39 25 32)(19 40 26 33)(20 41 27 34)(21 42 28 35)(57 101 64 108)(58 102 65 109)(59 103 66 110)(60 104 67 111)(61 105 68 112)(62 106 69 99)(63 107 70 100)(71 93 78 86)(72 94 79 87)(73 95 80 88)(74 96 81 89)(75 97 82 90)(76 98 83 91)(77 85 84 92)
(1 85 40 64)(2 93 41 58)(3 87 42 66)(4 95 36 60)(5 89 37 68)(6 97 38 62)(7 91 39 70)(8 90 31 69)(9 98 32 63)(10 92 33 57)(11 86 34 65)(12 94 35 59)(13 88 29 67)(14 96 30 61)(15 104 50 80)(16 112 51 74)(17 106 52 82)(18 100 53 76)(19 108 54 84)(20 102 55 78)(21 110 56 72)(22 111 43 73)(23 105 44 81)(24 99 45 75)(25 107 46 83)(26 101 47 77)(27 109 48 71)(28 103 49 79)
(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 11)(9 10)(12 14)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 112)(95 111)(96 110)(97 109)(98 108)
G:=sub<Sym(112)| (1,47,10,54)(2,48,11,55)(3,49,12,56)(4,43,13,50)(5,44,14,51)(6,45,8,52)(7,46,9,53)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,101,64,108)(58,102,65,109)(59,103,66,110)(60,104,67,111)(61,105,68,112)(62,106,69,99)(63,107,70,100)(71,93,78,86)(72,94,79,87)(73,95,80,88)(74,96,81,89)(75,97,82,90)(76,98,83,91)(77,85,84,92), (1,85,40,64)(2,93,41,58)(3,87,42,66)(4,95,36,60)(5,89,37,68)(6,97,38,62)(7,91,39,70)(8,90,31,69)(9,98,32,63)(10,92,33,57)(11,86,34,65)(12,94,35,59)(13,88,29,67)(14,96,30,61)(15,104,50,80)(16,112,51,74)(17,106,52,82)(18,100,53,76)(19,108,54,84)(20,102,55,78)(21,110,56,72)(22,111,43,73)(23,105,44,81)(24,99,45,75)(25,107,46,83)(26,101,47,77)(27,109,48,71)(28,103,49,79), (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)(8,11)(9,10)(12,14)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,112)(95,111)(96,110)(97,109)(98,108)>;
G:=Group( (1,47,10,54)(2,48,11,55)(3,49,12,56)(4,43,13,50)(5,44,14,51)(6,45,8,52)(7,46,9,53)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,101,64,108)(58,102,65,109)(59,103,66,110)(60,104,67,111)(61,105,68,112)(62,106,69,99)(63,107,70,100)(71,93,78,86)(72,94,79,87)(73,95,80,88)(74,96,81,89)(75,97,82,90)(76,98,83,91)(77,85,84,92), (1,85,40,64)(2,93,41,58)(3,87,42,66)(4,95,36,60)(5,89,37,68)(6,97,38,62)(7,91,39,70)(8,90,31,69)(9,98,32,63)(10,92,33,57)(11,86,34,65)(12,94,35,59)(13,88,29,67)(14,96,30,61)(15,104,50,80)(16,112,51,74)(17,106,52,82)(18,100,53,76)(19,108,54,84)(20,102,55,78)(21,110,56,72)(22,111,43,73)(23,105,44,81)(24,99,45,75)(25,107,46,83)(26,101,47,77)(27,109,48,71)(28,103,49,79), (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)(8,11)(9,10)(12,14)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,112)(95,111)(96,110)(97,109)(98,108) );
G=PermutationGroup([[(1,47,10,54),(2,48,11,55),(3,49,12,56),(4,43,13,50),(5,44,14,51),(6,45,8,52),(7,46,9,53),(15,36,22,29),(16,37,23,30),(17,38,24,31),(18,39,25,32),(19,40,26,33),(20,41,27,34),(21,42,28,35),(57,101,64,108),(58,102,65,109),(59,103,66,110),(60,104,67,111),(61,105,68,112),(62,106,69,99),(63,107,70,100),(71,93,78,86),(72,94,79,87),(73,95,80,88),(74,96,81,89),(75,97,82,90),(76,98,83,91),(77,85,84,92)], [(1,85,40,64),(2,93,41,58),(3,87,42,66),(4,95,36,60),(5,89,37,68),(6,97,38,62),(7,91,39,70),(8,90,31,69),(9,98,32,63),(10,92,33,57),(11,86,34,65),(12,94,35,59),(13,88,29,67),(14,96,30,61),(15,104,50,80),(16,112,51,74),(17,106,52,82),(18,100,53,76),(19,108,54,84),(20,102,55,78),(21,110,56,72),(22,111,43,73),(23,105,44,81),(24,99,45,75),(25,107,46,83),(26,101,47,77),(27,109,48,71),(28,103,49,79)], [(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),(8,11),(9,10),(12,14),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(43,50),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,112),(95,111),(96,110),(97,109),(98,108)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4xD7 | D28 | C7:D4 | C7:D4 | C8:C22 | D4:D14 |
| kernel | C4:C4:36D14 | C14.D8 | C2xC4.Dic7 | C7xC42:C2 | C22xD28 | C2xD28 | C2xC28 | C22xC14 | C42:C2 | C4:C4 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 6 | 3 | 12 | 12 | 6 | 6 | 2 | 12 |
Matrix representation of C4:C4:36D14 ►in GL6(F113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 38 | 0 | 0 |
| 0 | 0 | 93 | 9 | 0 | 0 |
| 0 | 0 | 109 | 36 | 58 | 75 |
| 0 | 0 | 77 | 0 | 38 | 55 |
| 3 | 41 | 0 | 0 | 0 | 0 |
| 110 | 110 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 46 | 0 | 70 |
| 0 | 0 | 21 | 84 | 43 | 58 |
| 0 | 0 | 80 | 71 | 9 | 67 |
| 0 | 0 | 42 | 0 | 46 | 104 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 33 | 0 | 0 |
| 0 | 0 | 54 | 112 | 0 | 0 |
| 0 | 0 | 89 | 45 | 80 | 80 |
| 0 | 0 | 13 | 94 | 33 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 66 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 79 | 9 | 0 | 0 |
| 0 | 0 | 60 | 34 | 0 | 0 |
| 0 | 0 | 99 | 41 | 4 | 109 |
| 0 | 0 | 42 | 19 | 32 | 109 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,104,93,109,77,0,0,38,9,36,0,0,0,0,0,58,38,0,0,0,0,75,55],[3,110,0,0,0,0,41,110,0,0,0,0,0,0,29,21,80,42,0,0,46,84,71,0,0,0,0,43,9,46,0,0,70,58,67,104],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,25,54,89,13,0,0,33,112,45,94,0,0,0,0,80,33,0,0,0,0,80,9],[1,66,0,0,0,0,0,112,0,0,0,0,0,0,79,60,99,42,0,0,9,34,41,19,0,0,0,0,4,32,0,0,0,0,109,109] >;
C4:C4:36D14 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{36}D_{14} % in TeX
G:=Group("C4:C4:36D14"); // GroupNames label
G:=SmallGroup(448,535);
// by ID
G=gap.SmallGroup(448,535);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,387,58,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations