metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:21D14, C14.732+ 1+4, (C2xQ8):9D14, (C4xC28):33C22, C22:C4:35D14, C4.4D4:13D7, D14:3Q8:31C2, C23:D14.6C2, (C2xD4).111D14, C42:D7:36C2, C4:Dic7:42C22, (Q8xC14):15C22, Dic7:4D4:32C2, D14.23(C4oD4), D14.D4:44C2, (C2xC28).632C23, (C2xC14).223C24, Dic7:C4:67C22, (C4xDic7):57C22, C2.76(D4:6D14), C23.D7:34C22, C23.45(C22xD7), D14:C4.136C22, C7:8(C22.45C24), (D4xC14).211C22, C23.D14:40C2, (C22xC14).53C23, (C23xD7).66C22, C22.244(C23xD7), C23.18D14:25C2, (C2xDic7).255C23, (C22xDic7):28C22, (C22xD7).217C23, C2.79(D7xC4oD4), (D7xC22:C4):19C2, C14.190(C2xC4oD4), (C7xC4.4D4):15C2, (C2xC4xD7).215C22, (C2xC4).74(C22xD7), (C7xC22:C4):31C22, (C2xC7:D4).61C22, SmallGroup(448,1132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:21D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1228 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C2xC22:C4, C42:C2, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xD7, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C7xQ8, C22xD7, C22xD7, C22xC14, C22.45C24, C4xDic7, Dic7:C4, C4:Dic7, D14:C4, C23.D7, C23.D7, C4xC28, C7xC22:C4, C2xC4xD7, C22xDic7, C2xC7:D4, D4xC14, Q8xC14, C23xD7, C42:D7, C23.D14, D7xC22:C4, Dic7:4D4, D14.D4, C23.18D14, C23:D14, D14:3Q8, C7xC4.4D4, C42:21D14
Quotients: C1, C2, C22, C23, D7, C4oD4, C24, D14, C2xC4oD4, 2+ 1+4, C22xD7, C22.45C24, C23xD7, D4:6D14, D7xC4oD4, C42:21D14
►On 112 points
Generators in S112
(1 80 17 35)(2 74 18 29)(3 82 19 37)(4 76 20 31)(5 84 21 39)(6 78 15 33)(7 72 16 41)(8 83 25 38)(9 77 26 32)(10 71 27 40)(11 79 28 34)(12 73 22 42)(13 81 23 36)(14 75 24 30)(43 90 108 97)(44 58 109 65)(45 92 110 85)(46 60 111 67)(47 94 112 87)(48 62 99 69)(49 96 100 89)(50 64 101 57)(51 98 102 91)(52 66 103 59)(53 86 104 93)(54 68 105 61)(55 88 106 95)(56 70 107 63)
(1 57 12 90)(2 65 13 98)(3 59 14 92)(4 67 8 86)(5 61 9 94)(6 69 10 88)(7 63 11 96)(15 62 27 95)(16 70 28 89)(17 64 22 97)(18 58 23 91)(19 66 24 85)(20 60 25 93)(21 68 26 87)(29 109 36 51)(30 45 37 103)(31 111 38 53)(32 47 39 105)(33 99 40 55)(34 49 41 107)(35 101 42 43)(44 81 102 74)(46 83 104 76)(48 71 106 78)(50 73 108 80)(52 75 110 82)(54 77 112 84)(56 79 100 72)
(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)(9 14)(10 13)(11 12)(15 18)(16 17)(19 21)(22 28)(23 27)(24 26)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 100)(44 99)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 81)(72 80)(73 79)(74 78)(75 77)(82 84)(85 94)(86 93)(87 92)(88 91)(89 90)(95 98)(96 97)
G:=sub<Sym(112)| (1,80,17,35)(2,74,18,29)(3,82,19,37)(4,76,20,31)(5,84,21,39)(6,78,15,33)(7,72,16,41)(8,83,25,38)(9,77,26,32)(10,71,27,40)(11,79,28,34)(12,73,22,42)(13,81,23,36)(14,75,24,30)(43,90,108,97)(44,58,109,65)(45,92,110,85)(46,60,111,67)(47,94,112,87)(48,62,99,69)(49,96,100,89)(50,64,101,57)(51,98,102,91)(52,66,103,59)(53,86,104,93)(54,68,105,61)(55,88,106,95)(56,70,107,63), (1,57,12,90)(2,65,13,98)(3,59,14,92)(4,67,8,86)(5,61,9,94)(6,69,10,88)(7,63,11,96)(15,62,27,95)(16,70,28,89)(17,64,22,97)(18,58,23,91)(19,66,24,85)(20,60,25,93)(21,68,26,87)(29,109,36,51)(30,45,37,103)(31,111,38,53)(32,47,39,105)(33,99,40,55)(34,49,41,107)(35,101,42,43)(44,81,102,74)(46,83,104,76)(48,71,106,78)(50,73,108,80)(52,75,110,82)(54,77,112,84)(56,79,100,72), (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)(9,14)(10,13)(11,12)(15,18)(16,17)(19,21)(22,28)(23,27)(24,26)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,100)(44,99)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84)(85,94)(86,93)(87,92)(88,91)(89,90)(95,98)(96,97)>;
G:=Group( (1,80,17,35)(2,74,18,29)(3,82,19,37)(4,76,20,31)(5,84,21,39)(6,78,15,33)(7,72,16,41)(8,83,25,38)(9,77,26,32)(10,71,27,40)(11,79,28,34)(12,73,22,42)(13,81,23,36)(14,75,24,30)(43,90,108,97)(44,58,109,65)(45,92,110,85)(46,60,111,67)(47,94,112,87)(48,62,99,69)(49,96,100,89)(50,64,101,57)(51,98,102,91)(52,66,103,59)(53,86,104,93)(54,68,105,61)(55,88,106,95)(56,70,107,63), (1,57,12,90)(2,65,13,98)(3,59,14,92)(4,67,8,86)(5,61,9,94)(6,69,10,88)(7,63,11,96)(15,62,27,95)(16,70,28,89)(17,64,22,97)(18,58,23,91)(19,66,24,85)(20,60,25,93)(21,68,26,87)(29,109,36,51)(30,45,37,103)(31,111,38,53)(32,47,39,105)(33,99,40,55)(34,49,41,107)(35,101,42,43)(44,81,102,74)(46,83,104,76)(48,71,106,78)(50,73,108,80)(52,75,110,82)(54,77,112,84)(56,79,100,72), (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)(9,14)(10,13)(11,12)(15,18)(16,17)(19,21)(22,28)(23,27)(24,26)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,100)(44,99)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84)(85,94)(86,93)(87,92)(88,91)(89,90)(95,98)(96,97) );
G=PermutationGroup([[(1,80,17,35),(2,74,18,29),(3,82,19,37),(4,76,20,31),(5,84,21,39),(6,78,15,33),(7,72,16,41),(8,83,25,38),(9,77,26,32),(10,71,27,40),(11,79,28,34),(12,73,22,42),(13,81,23,36),(14,75,24,30),(43,90,108,97),(44,58,109,65),(45,92,110,85),(46,60,111,67),(47,94,112,87),(48,62,99,69),(49,96,100,89),(50,64,101,57),(51,98,102,91),(52,66,103,59),(53,86,104,93),(54,68,105,61),(55,88,106,95),(56,70,107,63)], [(1,57,12,90),(2,65,13,98),(3,59,14,92),(4,67,8,86),(5,61,9,94),(6,69,10,88),(7,63,11,96),(15,62,27,95),(16,70,28,89),(17,64,22,97),(18,58,23,91),(19,66,24,85),(20,60,25,93),(21,68,26,87),(29,109,36,51),(30,45,37,103),(31,111,38,53),(32,47,39,105),(33,99,40,55),(34,49,41,107),(35,101,42,43),(44,81,102,74),(46,83,104,76),(48,71,106,78),(50,73,108,80),(52,75,110,82),(54,77,112,84),(56,79,100,72)], [(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),(9,14),(10,13),(11,12),(15,18),(16,17),(19,21),(22,28),(23,27),(24,26),(29,33),(30,32),(34,42),(35,41),(36,40),(37,39),(43,100),(44,99),(45,112),(46,111),(47,110),(48,109),(49,108),(50,107),(51,106),(52,105),(53,104),(54,103),(55,102),(56,101),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(71,81),(72,80),(73,79),(74,78),(75,77),(82,84),(85,94),(86,93),(87,92),(88,91),(89,90),(95,98),(96,97)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4oD4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4:6D14 | D7xC4oD4 |
| kernel | C42:21D14 | C42:D7 | C23.D14 | D7xC22:C4 | Dic7:4D4 | D14.D4 | C23.18D14 | C23:D14 | D14:3Q8 | C7xC4.4D4 | C4.4D4 | D14 | C42 | C22:C4 | C2xD4 | C2xQ8 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 8 | 3 | 12 | 3 | 3 | 1 | 6 | 12 |
Matrix representation of C42:21D14 ►in GL6(F29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 28 | 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 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 26 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,27,1],[1,0,0,0,0,0,28,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,10,12,0,0,0,0,8,1,0,0,0,0,0,0,1,28,0,0,0,0,0,28],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,22,16,0,0,0,0,26,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42:21D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_{14} % in TeX
G:=Group("C4^2:21D14"); // GroupNames label
G:=SmallGroup(448,1132);
// by ID
G=gap.SmallGroup(448,1132);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,387,100,346,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations