metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊7D14, C14.932+ 1+4, C4⋊C4⋊54D14, (C4×D28)⋊5C2, D28⋊26(C2×C4), (C2×D28)⋊18C4, (C4×C28)⋊4C22, C42⋊C2⋊6D7, D28⋊C4⋊11C2, D14⋊C4⋊60C22, (C2×C14).65C24, C14.17(C23×C4), C4⋊Dic7⋊82C22, D14.4(C22×C4), C2.1(D4⋊8D14), C28.120(C22×C4), (C2×C28).583C23, C22⋊C4.125D14, C7⋊2(C22.11C24), (C4×Dic7)⋊10C22, (C22×D28).17C2, (C22×C4).188D14, C22.27(C23×D7), (C2×D28).255C22, (C23×D7).35C22, C23.153(C22×D7), C23.D7.94C22, C23.21D14⋊24C2, (C22×C14).135C23, (C22×C28).225C22, (C2×Dic7).195C23, (C22×D7).162C23, (C2×C4)⋊6(C4×D7), C4.58(C2×C4×D7), (C2×C28)⋊11(C2×C4), (C2×C4×D7)⋊43C22, C22.27(C2×C4×D7), C2.19(D7×C22×C4), (C7×C4⋊C4)⋊51C22, (D7×C22⋊C4)⋊25C2, (C22×D7)⋊7(C2×C4), (C7×C42⋊C2)⋊7C2, (C2×C14).21(C22×C4), (C2×C4).271(C22×D7), (C7×C22⋊C4).135C22, SmallGroup(448,974)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1780 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22×D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22.11C24, C4×Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C22×C28, C23×D7, C4×D28, D7×C22⋊C4, D28⋊C4, C23.21D14, C7×C42⋊C2, C22×D28, C42⋊7D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C23×C4, 2+ 1+4, C4×D7, C22×D7, C22.11C24, C2×C4×D7, C23×D7, D7×C22×C4, D4⋊8D14, C42⋊7D14
►On 112 points
(1 35 56 9)(2 29 50 10)(3 30 51 11)(4 31 52 12)(5 32 53 13)(6 33 54 14)(7 34 55 8)(15 47 22 41)(16 48 23 42)(17 49 24 36)(18 43 25 37)(19 44 26 38)(20 45 27 39)(21 46 28 40)(57 108 64 101)(58 109 65 102)(59 110 66 103)(60 111 67 104)(61 112 68 105)(62 99 69 106)(63 100 70 107)(71 92 78 85)(72 93 79 86)(73 94 80 87)(74 95 81 88)(75 96 82 89)(76 97 83 90)(77 98 84 91)
(1 83 42 62)(2 77 36 70)(3 71 37 64)(4 79 38 58)(5 73 39 66)(6 81 40 60)(7 75 41 68)(8 89 22 112)(9 97 23 106)(10 91 24 100)(11 85 25 108)(12 93 26 102)(13 87 27 110)(14 95 28 104)(15 105 34 96)(16 99 35 90)(17 107 29 98)(18 101 30 92)(19 109 31 86)(20 103 32 94)(21 111 33 88)(43 57 51 78)(44 65 52 72)(45 59 53 80)(46 67 54 74)(47 61 55 82)(48 69 56 76)(49 63 50 84)
(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 54)(2 53)(3 52)(4 51)(5 50)(6 56)(7 55)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 32)(30 31)(33 35)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 65)(58 64)(59 63)(60 62)(66 70)(67 69)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(85 86)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(99 104)(100 103)(101 102)(105 112)(106 111)(107 110)(108 109)
G:=sub<Sym(112)| (1,35,56,9)(2,29,50,10)(3,30,51,11)(4,31,52,12)(5,32,53,13)(6,33,54,14)(7,34,55,8)(15,47,22,41)(16,48,23,42)(17,49,24,36)(18,43,25,37)(19,44,26,38)(20,45,27,39)(21,46,28,40)(57,108,64,101)(58,109,65,102)(59,110,66,103)(60,111,67,104)(61,112,68,105)(62,99,69,106)(63,100,70,107)(71,92,78,85)(72,93,79,86)(73,94,80,87)(74,95,81,88)(75,96,82,89)(76,97,83,90)(77,98,84,91), (1,83,42,62)(2,77,36,70)(3,71,37,64)(4,79,38,58)(5,73,39,66)(6,81,40,60)(7,75,41,68)(8,89,22,112)(9,97,23,106)(10,91,24,100)(11,85,25,108)(12,93,26,102)(13,87,27,110)(14,95,28,104)(15,105,34,96)(16,99,35,90)(17,107,29,98)(18,101,30,92)(19,109,31,86)(20,103,32,94)(21,111,33,88)(43,57,51,78)(44,65,52,72)(45,59,53,80)(46,67,54,74)(47,61,55,82)(48,69,56,76)(49,63,50,84), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,56)(7,55)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,32)(30,31)(33,35)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,65)(58,64)(59,63)(60,62)(66,70)(67,69)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,86)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(99,104)(100,103)(101,102)(105,112)(106,111)(107,110)(108,109)>;
G:=Group( (1,35,56,9)(2,29,50,10)(3,30,51,11)(4,31,52,12)(5,32,53,13)(6,33,54,14)(7,34,55,8)(15,47,22,41)(16,48,23,42)(17,49,24,36)(18,43,25,37)(19,44,26,38)(20,45,27,39)(21,46,28,40)(57,108,64,101)(58,109,65,102)(59,110,66,103)(60,111,67,104)(61,112,68,105)(62,99,69,106)(63,100,70,107)(71,92,78,85)(72,93,79,86)(73,94,80,87)(74,95,81,88)(75,96,82,89)(76,97,83,90)(77,98,84,91), (1,83,42,62)(2,77,36,70)(3,71,37,64)(4,79,38,58)(5,73,39,66)(6,81,40,60)(7,75,41,68)(8,89,22,112)(9,97,23,106)(10,91,24,100)(11,85,25,108)(12,93,26,102)(13,87,27,110)(14,95,28,104)(15,105,34,96)(16,99,35,90)(17,107,29,98)(18,101,30,92)(19,109,31,86)(20,103,32,94)(21,111,33,88)(43,57,51,78)(44,65,52,72)(45,59,53,80)(46,67,54,74)(47,61,55,82)(48,69,56,76)(49,63,50,84), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,56)(7,55)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,32)(30,31)(33,35)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,65)(58,64)(59,63)(60,62)(66,70)(67,69)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,86)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(99,104)(100,103)(101,102)(105,112)(106,111)(107,110)(108,109) );
G=PermutationGroup([[(1,35,56,9),(2,29,50,10),(3,30,51,11),(4,31,52,12),(5,32,53,13),(6,33,54,14),(7,34,55,8),(15,47,22,41),(16,48,23,42),(17,49,24,36),(18,43,25,37),(19,44,26,38),(20,45,27,39),(21,46,28,40),(57,108,64,101),(58,109,65,102),(59,110,66,103),(60,111,67,104),(61,112,68,105),(62,99,69,106),(63,100,70,107),(71,92,78,85),(72,93,79,86),(73,94,80,87),(74,95,81,88),(75,96,82,89),(76,97,83,90),(77,98,84,91)], [(1,83,42,62),(2,77,36,70),(3,71,37,64),(4,79,38,58),(5,73,39,66),(6,81,40,60),(7,75,41,68),(8,89,22,112),(9,97,23,106),(10,91,24,100),(11,85,25,108),(12,93,26,102),(13,87,27,110),(14,95,28,104),(15,105,34,96),(16,99,35,90),(17,107,29,98),(18,101,30,92),(19,109,31,86),(20,103,32,94),(21,111,33,88),(43,57,51,78),(44,65,52,72),(45,59,53,80),(46,67,54,74),(47,61,55,82),(48,69,56,76),(49,63,50,84)], [(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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,56),(7,55),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,32),(30,31),(33,35),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46),(57,65),(58,64),(59,63),(60,62),(66,70),(67,69),(71,79),(72,78),(73,77),(74,76),(80,84),(81,83),(85,86),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(99,104),(100,103),(101,102),(105,112),(106,111),(107,110),(108,109)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2M | 4A | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | D14 | D14 | D14 | D14 | C4×D7 | 2+ 1+4 | D4⋊8D14 |
| kernel | C42⋊7D14 | C4×D28 | D7×C22⋊C4 | D28⋊C4 | C23.21D14 | C7×C42⋊C2 | C22×D28 | C2×D28 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C14 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 3 | 6 | 6 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of C42⋊7D14 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 24 | 0 | 0 |
| 0 | 0 | 5 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 0 | 5 | 13 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 19 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 4 | 4 | 4 | 4 |
| 0 | 0 | 25 | 18 | 25 | 18 |
| 26 | 10 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 4 | 4 | 4 | 4 |
| 0 | 0 | 18 | 25 | 18 | 25 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,5,0,0,0,0,24,13,0,0,0,0,0,0,16,5,0,0,0,0,24,13],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,27,0,1,0,0,0,0,27,0,1],[0,3,0,0,0,0,19,26,0,0,0,0,0,0,25,4,4,25,0,0,25,11,4,18,0,0,0,0,4,25,0,0,0,0,4,18],[26,5,0,0,0,0,10,3,0,0,0,0,0,0,25,11,4,18,0,0,25,4,4,25,0,0,0,0,4,18,0,0,0,0,4,25] >;
C42⋊7D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7D_{14} % in TeX
G:=Group("C4^2:7D14"); // GroupNames label
G:=SmallGroup(448,974);
// by ID
G=gap.SmallGroup(448,974);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations