direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊2D7, D4⋊5D14, C14.6C24, C28.20C23, D14.2C23, C23.19D14, Dic14⋊7C22, Dic7.3C23, (C2×D4)⋊8D7, (D4×C14)⋊6C2, C14⋊2(C4○D4), (C2×C4).60D14, (C7×D4)⋊6C22, (C4×D7)⋊4C22, C7⋊D4⋊2C22, C2.7(C23×D7), (C2×C14).1C23, C4.20(C22×D7), (C2×Dic14)⋊12C2, (C2×C28).45C22, (C22×Dic7)⋊8C2, (C2×Dic7)⋊9C22, C22.1(C22×D7), (C22×C14).23C22, (C22×D7).29C22, (C2×C4×D7)⋊4C2, C7⋊2(C2×C4○D4), (C2×C7⋊D4)⋊10C2, SmallGroup(224,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊2D7
G = < a,b,c,d,e | a2=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 542 in 164 conjugacy classes, 89 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C14, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C2×D4⋊2D7
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C22×D7, D4⋊2D7, C23×D7, C2×D4⋊2D7
►On 112 points
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
(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 14)(9 13)(10 12)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)
G:=sub<Sym(112)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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)(8,14)(9,13)(10,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,35)(30,34)(31,33)(36,42)(37,41)(38,40)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98), (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)(8,14)(9,13)(10,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,35)(30,34)(31,33)(36,42)(37,41)(38,40)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106) );
G=PermutationGroup([[(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,106,78,99),(72,107,79,100),(73,108,80,101),(74,109,81,102),(75,110,82,103),(76,111,83,104),(77,112,84,105)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98)], [(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),(8,14),(9,13),(10,12),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,35),(30,34),(31,33),(36,42),(37,41),(38,40),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(57,63),(58,62),(59,61),(64,70),(65,69),(66,68),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(85,91),(86,90),(87,89),(92,98),(93,97),(94,96),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106)]])
C2×D4⋊2D7 is a maximal subgroup of
C23⋊C4⋊5D7 M4(2).19D14 D4⋊(C4×D7) D4⋊2D7⋊C4 D4⋊3D28 D4.D28 Dic14⋊D4 Dic14.16D4 C42.108D14 D4⋊5D28 D4⋊6D28 C24.56D14 C24.33D14 C24.34D14 C28⋊(C4○D4) C14.682- 1+4 Dic14⋊19D4 Dic14⋊20D4 C4⋊C4⋊21D14 C14.722- 1+4 C14.402+ 1+4 C14.732- 1+4 C14.792- 1+4 C14.822- 1+4 C4⋊C4⋊28D14 C42.233D14 C42.141D14 Dic14⋊10D4 C42⋊26D14 C42.238D14 Dic14⋊11D4 SD16⋊D14 C24.42D14 C14.1042- 1+4 C2×D7×C4○D4 D14.C24
C2×D4⋊2D7 is a maximal quotient of
C24.31D14 C14.52- 1+4 C42.102D14 C42.105D14 C42.106D14 D4⋊6Dic14 D4⋊6D28 C42.229D14 C42.117D14 C42.119D14 C24.56D14 C24.32D14 C24.33D14 C24.35D14 C28⋊(C4○D4) Dic14⋊19D4 C4⋊C4.178D14 C14.342+ 1+4 C14.352+ 1+4 C14.712- 1+4 C4⋊C4⋊21D14 C14.732- 1+4 C14.432+ 1+4 C14.452+ 1+4 C14.462+ 1+4 C14.1152+ 1+4 C14.472+ 1+4 (Q8×Dic7)⋊C2 C22⋊Q8⋊25D7 C14.152- 1+4 C14.1182+ 1+4 C14.212- 1+4 C14.232- 1+4 C14.772- 1+4 C14.242- 1+4 C4⋊C4.197D14 C14.802- 1+4 C14.1222+ 1+4 C14.852- 1+4 C42.139D14 C42.234D14 C42.143D14 C42.144D14 C42.166D14 C42.238D14 Dic14⋊11D4 C42.168D14 Dic14⋊8Q8 C42.241D14 C42.176D14 C42.177D14 C2×D4×Dic7 C24.42D14
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D4⋊2D7 |
| kernel | C2×D4⋊2D7 | C2×Dic14 | C2×C4×D7 | D4⋊2D7 | C22×Dic7 | C2×C7⋊D4 | D4×C14 | C2×D4 | C14 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 3 | 4 | 3 | 12 | 6 | 6 |
Matrix representation of C2×D4⋊2D7 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 16 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 3 |
| 1 | 0 | 0 | 0 |
| 16 | 28 | 0 | 0 |
| 0 | 0 | 26 | 28 |
| 0 | 0 | 8 | 3 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[17,11,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[17,11,0,0,16,12,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,28,3],[1,16,0,0,0,28,0,0,0,0,26,8,0,0,28,3] >;
C2×D4⋊2D7 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_2D_7
% in TeX
G:=Group("C2xD4:2D7"); // GroupNames label
G:=SmallGroup(224,179);
// by ID
G=gap.SmallGroup(224,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,579,159,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations