metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊22D14, C14.1292+ 1+4, (C2×Q8)⋊10D14, (C4×C28)⋊29C22, D14⋊C4⋊6C22, C22⋊C4⋊21D14, C4.4D4⋊16D7, C22⋊D28⋊26C2, C23⋊D14⋊25C2, (C2×D4).112D14, C4.D28⋊30C2, (C2×C28).83C23, (Q8×C14)⋊16C22, C28.23D4⋊24C2, (C2×C14).227C24, C7⋊2(C24⋊C22), (C4×Dic7)⋊37C22, (C2×D28).34C22, (C23×D7)⋊12C22, C23.D7⋊35C22, C2.77(D4⋊6D14), C2.53(D4⋊8D14), C23.49(C22×D7), Dic7.D4⋊43C2, (C2×Dic14)⋊10C22, (D4×C14).212C22, (C22×C14).57C23, (C22×D7).99C23, C22.248(C23×D7), (C2×Dic7).117C23, (C7×C4.4D4)⋊19C2, (C7×C22⋊C4)⋊32C22, (C2×C4).200(C22×D7), (C2×C7⋊D4).65C22, SmallGroup(448,1136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊22D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1676 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C22≀C2, C4.4D4, C4.4D4, Dic14, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C24⋊C22, C4×Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×Dic14, C2×D28, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C4.D28, C22⋊D28, Dic7.D4, C23⋊D14, C28.23D4, C7×C4.4D4, C42⋊22D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C24⋊C22, C23×D7, D4⋊6D14, D4⋊8D14, C42⋊22D14
►On 112 points
Generators in S112
(1 8 15 22)(2 23 16 9)(3 10 17 24)(4 25 18 11)(5 12 19 26)(6 27 20 13)(7 14 21 28)(29 67 76 53)(30 54 77 68)(31 69 78 55)(32 56 79 70)(33 57 80 43)(34 44 81 58)(35 59 82 45)(36 46 83 60)(37 61 84 47)(38 48 71 62)(39 63 72 49)(40 50 73 64)(41 65 74 51)(42 52 75 66)(85 101 108 92)(86 93 109 102)(87 103 110 94)(88 95 111 104)(89 105 112 96)(90 97 99 106)(91 107 100 98)
(1 31 88 48)(2 63 89 79)(3 33 90 50)(4 65 91 81)(5 35 92 52)(6 67 93 83)(7 37 94 54)(8 69 95 71)(9 39 96 56)(10 57 97 73)(11 41 98 44)(12 59 85 75)(13 29 86 46)(14 61 87 77)(15 78 111 62)(16 49 112 32)(17 80 99 64)(18 51 100 34)(19 82 101 66)(20 53 102 36)(21 84 103 68)(22 55 104 38)(23 72 105 70)(24 43 106 40)(25 74 107 58)(26 45 108 42)(27 76 109 60)(28 47 110 30)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 61)(16 60)(17 59)(18 58)(19 57)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 89)(30 88)(31 87)(32 86)(33 85)(34 98)(35 97)(36 96)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(71 103)(72 102)(73 101)(74 100)(75 99)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)
G:=sub<Sym(112)| (1,8,15,22)(2,23,16,9)(3,10,17,24)(4,25,18,11)(5,12,19,26)(6,27,20,13)(7,14,21,28)(29,67,76,53)(30,54,77,68)(31,69,78,55)(32,56,79,70)(33,57,80,43)(34,44,81,58)(35,59,82,45)(36,46,83,60)(37,61,84,47)(38,48,71,62)(39,63,72,49)(40,50,73,64)(41,65,74,51)(42,52,75,66)(85,101,108,92)(86,93,109,102)(87,103,110,94)(88,95,111,104)(89,105,112,96)(90,97,99,106)(91,107,100,98), (1,31,88,48)(2,63,89,79)(3,33,90,50)(4,65,91,81)(5,35,92,52)(6,67,93,83)(7,37,94,54)(8,69,95,71)(9,39,96,56)(10,57,97,73)(11,41,98,44)(12,59,85,75)(13,29,86,46)(14,61,87,77)(15,78,111,62)(16,49,112,32)(17,80,99,64)(18,51,100,34)(19,82,101,66)(20,53,102,36)(21,84,103,68)(22,55,104,38)(23,72,105,70)(24,43,106,40)(25,74,107,58)(26,45,108,42)(27,76,109,60)(28,47,110,30), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,61)(16,60)(17,59)(18,58)(19,57)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,89)(30,88)(31,87)(32,86)(33,85)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(71,103)(72,102)(73,101)(74,100)(75,99)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)>;
G:=Group( (1,8,15,22)(2,23,16,9)(3,10,17,24)(4,25,18,11)(5,12,19,26)(6,27,20,13)(7,14,21,28)(29,67,76,53)(30,54,77,68)(31,69,78,55)(32,56,79,70)(33,57,80,43)(34,44,81,58)(35,59,82,45)(36,46,83,60)(37,61,84,47)(38,48,71,62)(39,63,72,49)(40,50,73,64)(41,65,74,51)(42,52,75,66)(85,101,108,92)(86,93,109,102)(87,103,110,94)(88,95,111,104)(89,105,112,96)(90,97,99,106)(91,107,100,98), (1,31,88,48)(2,63,89,79)(3,33,90,50)(4,65,91,81)(5,35,92,52)(6,67,93,83)(7,37,94,54)(8,69,95,71)(9,39,96,56)(10,57,97,73)(11,41,98,44)(12,59,85,75)(13,29,86,46)(14,61,87,77)(15,78,111,62)(16,49,112,32)(17,80,99,64)(18,51,100,34)(19,82,101,66)(20,53,102,36)(21,84,103,68)(22,55,104,38)(23,72,105,70)(24,43,106,40)(25,74,107,58)(26,45,108,42)(27,76,109,60)(28,47,110,30), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,61)(16,60)(17,59)(18,58)(19,57)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,89)(30,88)(31,87)(32,86)(33,85)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(71,103)(72,102)(73,101)(74,100)(75,99)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104) );
G=PermutationGroup([[(1,8,15,22),(2,23,16,9),(3,10,17,24),(4,25,18,11),(5,12,19,26),(6,27,20,13),(7,14,21,28),(29,67,76,53),(30,54,77,68),(31,69,78,55),(32,56,79,70),(33,57,80,43),(34,44,81,58),(35,59,82,45),(36,46,83,60),(37,61,84,47),(38,48,71,62),(39,63,72,49),(40,50,73,64),(41,65,74,51),(42,52,75,66),(85,101,108,92),(86,93,109,102),(87,103,110,94),(88,95,111,104),(89,105,112,96),(90,97,99,106),(91,107,100,98)], [(1,31,88,48),(2,63,89,79),(3,33,90,50),(4,65,91,81),(5,35,92,52),(6,67,93,83),(7,37,94,54),(8,69,95,71),(9,39,96,56),(10,57,97,73),(11,41,98,44),(12,59,85,75),(13,29,86,46),(14,61,87,77),(15,78,111,62),(16,49,112,32),(17,80,99,64),(18,51,100,34),(19,82,101,66),(20,53,102,36),(21,84,103,68),(22,55,104,38),(23,72,105,70),(24,43,106,40),(25,74,107,58),(26,45,108,42),(27,76,109,60),(28,47,110,30)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,61),(16,60),(17,59),(18,58),(19,57),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,89),(30,88),(31,87),(32,86),(33,85),(34,98),(35,97),(36,96),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(71,103),(72,102),(73,101),(74,100),(75,99),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 | D4⋊8D14 |
| kernel | C42⋊22D14 | C4.D28 | C22⋊D28 | Dic7.D4 | C23⋊D14 | C28.23D4 | C7×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 3 | 3 | 12 | 3 | 3 | 3 | 6 | 12 |
Matrix representation of C42⋊22D14 ►in GL8(𝔽29)
| 1 | 0 | 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 1 | 21 | 26 | 0 | 0 | 0 | 0 |
| 22 | 25 | 28 | 0 | 0 | 0 | 0 | 0 |
| 9 | 21 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 | 4 | 14 |
| 0 | 0 | 0 | 0 | 21 | 28 | 1 | 25 |
| 27 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 27 | 0 |
| 0 | 0 | 0 | 0 | 21 | 19 | 0 | 27 |
| 0 | 0 | 0 | 0 | 20 | 28 | 0 | 23 |
| 0 | 0 | 0 | 0 | 11 | 12 | 8 | 10 |
| 21 | 4 | 12 | 24 | 0 | 0 | 0 | 0 |
| 5 | 1 | 8 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 7 | 14 | 8 |
| 0 | 0 | 0 | 0 | 22 | 21 | 13 | 26 |
| 24 | 7 | 15 | 16 | 0 | 0 | 0 | 0 |
| 9 | 5 | 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 17 | 5 | 18 |
| 0 | 0 | 0 | 0 | 9 | 4 | 0 | 24 |
| 0 | 0 | 0 | 0 | 4 | 22 | 0 | 14 |
| 0 | 0 | 0 | 0 | 15 | 22 | 9 | 11 |
G:=sub<GL(8,GF(29))| [1,0,22,9,0,0,0,0,0,1,25,21,0,0,0,0,9,21,28,0,0,0,0,0,10,26,0,28,0,0,0,0,0,0,0,0,4,1,1,21,0,0,0,0,14,25,17,28,0,0,0,0,0,0,4,1,0,0,0,0,0,0,14,25],[27,28,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,0,5,21,0,0,0,0,0,0,0,0,0,21,20,11,0,0,0,0,6,19,28,12,0,0,0,0,27,0,0,8,0,0,0,0,0,27,23,10],[21,5,0,0,0,0,0,0,4,1,0,0,0,0,0,0,12,8,0,22,0,0,0,0,24,18,25,7,0,0,0,0,0,0,0,0,15,16,22,22,0,0,0,0,21,3,7,21,0,0,0,0,0,0,14,13,0,0,0,0,0,0,8,26],[24,9,0,0,0,0,0,0,7,5,0,0,0,0,0,0,15,12,25,5,0,0,0,0,16,17,26,4,0,0,0,0,0,0,0,0,14,9,4,15,0,0,0,0,17,4,22,22,0,0,0,0,5,0,0,9,0,0,0,0,18,24,14,11] >;
C42⋊22D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{22}D_{14} % in TeX
G:=Group("C4^2:22D14"); // GroupNames label
G:=SmallGroup(448,1136);
// by ID
G=gap.SmallGroup(448,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,192,18822]);
// 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*b^2,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations