metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:12D14, (C4xD4):9D7, C4:C4:45D14, (C4xD7):14D4, (C4xD28):25C2, (D4xC28):11C2, C4.219(D4xD7), D14:2(C4oD4), (C4xC28):17C22, C22:C4:44D14, D14.38(C2xD4), C28.378(C2xD4), (C22xC4):10D14, C23:D14:33C2, C22:D28:30C2, D14:D4:47C2, D14:C4:65C22, D14:Q8:51C2, (C2xD4).211D14, C22:2(C4oD28), C42:D7:13C2, (C2xC14).91C24, C4:Dic7:57C22, Dic7.43(C2xD4), C14.47(C22xD4), Dic7:D4:46C2, D14.D4:53C2, D14.5D4:49C2, (C2xC28).157C23, Dic7:C4:70C22, (C22xC28):15C22, C7:2(C22.19C24), (C4xDic7):51C22, C23.D7:49C22, C22:Dic14:50C2, (C2xDic14):52C22, (C2xD28).209C22, (D4xC14).304C22, (C2xDic7).38C23, C23.170(C22xD7), C22.116(C23xD7), (C22xC14).161C23, (C23xD7).106C22, (C22xD7).170C23, (C22xDic7).220C22, C2.19(C2xD4xD7), (C2xC4oD28):5C2, (C4xC7:D4):42C2, C2.20(D7xC4oD4), (C2xC4xD7):47C22, (D7xC22xC4):22C2, (C2xC14):1(C4oD4), (C7xC4:C4):57C22, C2.43(C2xC4oD28), C14.39(C2xC4oD4), (C2xC7:D4):37C22, (C7xC22:C4):55C22, (C2xC4).156(C22xD7), SmallGroup(448,1000)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:12D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1620 in 330 conjugacy classes, 109 normal (91 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic7, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xC14, C42:C2, C4xD4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C23xC4, C2xC4oD4, Dic14, C4xD7, C4xD7, D28, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C22xD7, C22xD7, C22xC14, C22.19C24, C4xDic7, Dic7:C4, C4:Dic7, D14:C4, C23.D7, C4xC28, C7xC22:C4, C7xC4:C4, C2xDic14, C2xC4xD7, C2xC4xD7, C2xD28, C4oD28, C22xDic7, C2xC7:D4, C22xC28, D4xC14, C23xD7, C42:D7, C4xD28, C22:Dic14, C22:D28, D14.D4, D14:D4, D14.5D4, D14:Q8, C4xC7:D4, C23:D14, Dic7:D4, D4xC28, D7xC22xC4, C2xC4oD28, C42:12D14
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C4oD4, C24, D14, C22xD4, C2xC4oD4, C22xD7, C22.19C24, C4oD28, D4xD7, C23xD7, C2xC4oD28, C2xD4xD7, D7xC4oD4, C42:12D14
►On 112 points
(1 73 12 80)(2 81 13 74)(3 75 14 82)(4 83 8 76)(5 77 9 84)(6 71 10 78)(7 79 11 72)(15 63 31 70)(16 57 32 64)(17 65 33 58)(18 59 34 66)(19 67 35 60)(20 61 29 68)(21 69 30 62)(22 93 49 86)(23 87 43 94)(24 95 44 88)(25 89 45 96)(26 97 46 90)(27 91 47 98)(28 85 48 92)(36 104 53 111)(37 112 54 105)(38 106 55 99)(39 100 56 107)(40 108 50 101)(41 102 51 109)(42 110 52 103)
(1 39 46 19)(2 40 47 20)(3 41 48 21)(4 42 49 15)(5 36 43 16)(6 37 44 17)(7 38 45 18)(8 52 22 31)(9 53 23 32)(10 54 24 33)(11 55 25 34)(12 56 26 35)(13 50 27 29)(14 51 28 30)(57 77 104 94)(58 78 105 95)(59 79 106 96)(60 80 107 97)(61 81 108 98)(62 82 109 85)(63 83 110 86)(64 84 111 87)(65 71 112 88)(66 72 99 89)(67 73 100 90)(68 74 101 91)(69 75 102 92)(70 76 103 93)
(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)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 33)(30 32)(34 35)(36 41)(37 40)(38 39)(43 48)(44 47)(45 46)(50 54)(51 53)(55 56)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 112)(69 111)(70 110)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)
G:=sub<Sym(112)| (1,73,12,80)(2,81,13,74)(3,75,14,82)(4,83,8,76)(5,77,9,84)(6,71,10,78)(7,79,11,72)(15,63,31,70)(16,57,32,64)(17,65,33,58)(18,59,34,66)(19,67,35,60)(20,61,29,68)(21,69,30,62)(22,93,49,86)(23,87,43,94)(24,95,44,88)(25,89,45,96)(26,97,46,90)(27,91,47,98)(28,85,48,92)(36,104,53,111)(37,112,54,105)(38,106,55,99)(39,100,56,107)(40,108,50,101)(41,102,51,109)(42,110,52,103), (1,39,46,19)(2,40,47,20)(3,41,48,21)(4,42,49,15)(5,36,43,16)(6,37,44,17)(7,38,45,18)(8,52,22,31)(9,53,23,32)(10,54,24,33)(11,55,25,34)(12,56,26,35)(13,50,27,29)(14,51,28,30)(57,77,104,94)(58,78,105,95)(59,79,106,96)(60,80,107,97)(61,81,108,98)(62,82,109,85)(63,83,110,86)(64,84,111,87)(65,71,112,88)(66,72,99,89)(67,73,100,90)(68,74,101,91)(69,75,102,92)(70,76,103,93), (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)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,33)(30,32)(34,35)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46)(50,54)(51,53)(55,56)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,112)(69,111)(70,110)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)>;
G:=Group( (1,73,12,80)(2,81,13,74)(3,75,14,82)(4,83,8,76)(5,77,9,84)(6,71,10,78)(7,79,11,72)(15,63,31,70)(16,57,32,64)(17,65,33,58)(18,59,34,66)(19,67,35,60)(20,61,29,68)(21,69,30,62)(22,93,49,86)(23,87,43,94)(24,95,44,88)(25,89,45,96)(26,97,46,90)(27,91,47,98)(28,85,48,92)(36,104,53,111)(37,112,54,105)(38,106,55,99)(39,100,56,107)(40,108,50,101)(41,102,51,109)(42,110,52,103), (1,39,46,19)(2,40,47,20)(3,41,48,21)(4,42,49,15)(5,36,43,16)(6,37,44,17)(7,38,45,18)(8,52,22,31)(9,53,23,32)(10,54,24,33)(11,55,25,34)(12,56,26,35)(13,50,27,29)(14,51,28,30)(57,77,104,94)(58,78,105,95)(59,79,106,96)(60,80,107,97)(61,81,108,98)(62,82,109,85)(63,83,110,86)(64,84,111,87)(65,71,112,88)(66,72,99,89)(67,73,100,90)(68,74,101,91)(69,75,102,92)(70,76,103,93), (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)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,33)(30,32)(34,35)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46)(50,54)(51,53)(55,56)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,112)(69,111)(70,110)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92) );
G=PermutationGroup([[(1,73,12,80),(2,81,13,74),(3,75,14,82),(4,83,8,76),(5,77,9,84),(6,71,10,78),(7,79,11,72),(15,63,31,70),(16,57,32,64),(17,65,33,58),(18,59,34,66),(19,67,35,60),(20,61,29,68),(21,69,30,62),(22,93,49,86),(23,87,43,94),(24,95,44,88),(25,89,45,96),(26,97,46,90),(27,91,47,98),(28,85,48,92),(36,104,53,111),(37,112,54,105),(38,106,55,99),(39,100,56,107),(40,108,50,101),(41,102,51,109),(42,110,52,103)], [(1,39,46,19),(2,40,47,20),(3,41,48,21),(4,42,49,15),(5,36,43,16),(6,37,44,17),(7,38,45,18),(8,52,22,31),(9,53,23,32),(10,54,24,33),(11,55,25,34),(12,56,26,35),(13,50,27,29),(14,51,28,30),(57,77,104,94),(58,78,105,95),(59,79,106,96),(60,80,107,97),(61,81,108,98),(62,82,109,85),(63,83,110,86),(64,84,111,87),(65,71,112,88),(66,72,99,89),(67,73,100,90),(68,74,101,91),(69,75,102,92),(70,76,103,93)], [(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),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,33),(30,32),(34,35),(36,41),(37,40),(38,39),(43,48),(44,47),(45,46),(50,54),(51,53),(55,56),(57,109),(58,108),(59,107),(60,106),(61,105),(62,104),(63,103),(64,102),(65,101),(66,100),(67,99),(68,112),(69,111),(70,110),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4oD4 | C4oD4 | D14 | D14 | D14 | D14 | D14 | C4oD28 | D4xD7 | D7xC4oD4 |
| kernel | C42:12D14 | C42:D7 | C4xD28 | C22:Dic14 | C22:D28 | D14.D4 | D14:D4 | D14.5D4 | D14:Q8 | C4xC7:D4 | C23:D14 | Dic7:D4 | D4xC28 | D7xC22xC4 | C2xC4oD28 | C4xD7 | C4xD4 | D14 | C2xC14 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 6 | 6 |
Matrix representation of C42:12D14 ►in GL4(F29) generated by
| 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 26 |
| 0 | 0 | 16 | 22 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 27 | 13 |
| 0 | 0 | 8 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 7 |
| 0 | 0 | 3 | 26 |
G:=sub<GL(4,GF(29))| [0,1,0,0,28,0,0,0,0,0,7,16,0,0,26,22],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,28,0,0,0,0,27,8,0,0,13,20],[1,0,0,0,0,1,0,0,0,0,3,3,0,0,7,26] >;
C42:12D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{12}D_{14} % in TeX
G:=Group("C4^2:12D14"); // GroupNames label
G:=SmallGroup(448,1000);
// by ID
G=gap.SmallGroup(448,1000);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,675,570,18822]);
// 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^-1,d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations