direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4⋊1D4, C42⋊34D14, C4⋊1(D4×D7), (C4×D7)⋊7D4, C28⋊2(C2×D4), (C2×D4)⋊25D14, Dic7⋊1(C2×D4), C28⋊D4⋊25C2, C28⋊4D4⋊16C2, (C4×C28)⋊25C22, D14.61(C2×D4), (D7×C42)⋊12C2, (C2×D28)⋊30C22, (D4×C14)⋊17C22, C14.92(C22×D4), (C2×C28).507C23, (C2×C14).258C24, (C4×Dic7)⋊65C22, C23.64(C22×D7), (C22×C14).72C23, (C23×D7).71C22, C22.279(C23×D7), (C2×Dic7).268C23, (C22×D7).259C23, (C2×D4×D7)⋊18C2, C7⋊2(C2×C4⋊1D4), C2.65(C2×D4×D7), (C7×C4⋊1D4)⋊5C2, (C2×C7⋊D4)⋊25C22, (C2×C4×D7).250C22, (C2×C4).596(C22×D7), SmallGroup(448,1167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4⋊1D4
G = < a,b,c,d,e | a7=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 2988 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, D7, C14, C14, C42, C42, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C42, C4⋊1D4, C4⋊1D4, C22×D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4⋊1D4, C4×Dic7, C4×C28, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, D7×C42, C28⋊4D4, C28⋊D4, C7×C4⋊1D4, C2×D4×D7, D7×C4⋊1D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4⋊1D4, C22×D4, C22×D7, C2×C4⋊1D4, D4×D7, C23×D7, C2×D4×D7, D7×C4⋊1D4
►On 112 points
Generators in S112
(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 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)
(1 55 27 34)(2 56 28 35)(3 50 22 29)(4 51 23 30)(5 52 24 31)(6 53 25 32)(7 54 26 33)(8 43 15 36)(9 44 16 37)(10 45 17 38)(11 46 18 39)(12 47 19 40)(13 48 20 41)(14 49 21 42)(57 85 78 106)(58 86 79 107)(59 87 80 108)(60 88 81 109)(61 89 82 110)(62 90 83 111)(63 91 84 112)(64 92 71 99)(65 93 72 100)(66 94 73 101)(67 95 74 102)(68 96 75 103)(69 97 76 104)(70 98 77 105)
(1 90 20 104)(2 91 21 105)(3 85 15 99)(4 86 16 100)(5 87 17 101)(6 88 18 102)(7 89 19 103)(8 92 22 106)(9 93 23 107)(10 94 24 108)(11 95 25 109)(12 96 26 110)(13 97 27 111)(14 98 28 112)(29 57 43 71)(30 58 44 72)(31 59 45 73)(32 60 46 74)(33 61 47 75)(34 62 48 76)(35 63 49 77)(36 64 50 78)(37 65 51 79)(38 66 52 80)(39 67 53 81)(40 68 54 82)(41 69 55 83)(42 70 56 84)
(1 76)(2 77)(3 71)(4 72)(5 73)(6 74)(7 75)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)
G:=sub<Sym(112)| (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)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111), (1,55,27,34)(2,56,28,35)(3,50,22,29)(4,51,23,30)(5,52,24,31)(6,53,25,32)(7,54,26,33)(8,43,15,36)(9,44,16,37)(10,45,17,38)(11,46,18,39)(12,47,19,40)(13,48,20,41)(14,49,21,42)(57,85,78,106)(58,86,79,107)(59,87,80,108)(60,88,81,109)(61,89,82,110)(62,90,83,111)(63,91,84,112)(64,92,71,99)(65,93,72,100)(66,94,73,101)(67,95,74,102)(68,96,75,103)(69,97,76,104)(70,98,77,105), (1,90,20,104)(2,91,21,105)(3,85,15,99)(4,86,16,100)(5,87,17,101)(6,88,18,102)(7,89,19,103)(8,92,22,106)(9,93,23,107)(10,94,24,108)(11,95,25,109)(12,96,26,110)(13,97,27,111)(14,98,28,112)(29,57,43,71)(30,58,44,72)(31,59,45,73)(32,60,46,74)(33,61,47,75)(34,62,48,76)(35,63,49,77)(36,64,50,78)(37,65,51,79)(38,66,52,80)(39,67,53,81)(40,68,54,82)(41,69,55,83)(42,70,56,84), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)>;
G:=Group( (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)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111), (1,55,27,34)(2,56,28,35)(3,50,22,29)(4,51,23,30)(5,52,24,31)(6,53,25,32)(7,54,26,33)(8,43,15,36)(9,44,16,37)(10,45,17,38)(11,46,18,39)(12,47,19,40)(13,48,20,41)(14,49,21,42)(57,85,78,106)(58,86,79,107)(59,87,80,108)(60,88,81,109)(61,89,82,110)(62,90,83,111)(63,91,84,112)(64,92,71,99)(65,93,72,100)(66,94,73,101)(67,95,74,102)(68,96,75,103)(69,97,76,104)(70,98,77,105), (1,90,20,104)(2,91,21,105)(3,85,15,99)(4,86,16,100)(5,87,17,101)(6,88,18,102)(7,89,19,103)(8,92,22,106)(9,93,23,107)(10,94,24,108)(11,95,25,109)(12,96,26,110)(13,97,27,111)(14,98,28,112)(29,57,43,71)(30,58,44,72)(31,59,45,73)(32,60,46,74)(33,61,47,75)(34,62,48,76)(35,63,49,77)(36,64,50,78)(37,65,51,79)(38,66,52,80)(39,67,53,81)(40,68,54,82)(41,69,55,83)(42,70,56,84), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98) );
G=PermutationGroup([[(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),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83),(85,87),(88,91),(89,90),(92,94),(95,98),(96,97),(99,101),(102,105),(103,104),(106,108),(109,112),(110,111)], [(1,55,27,34),(2,56,28,35),(3,50,22,29),(4,51,23,30),(5,52,24,31),(6,53,25,32),(7,54,26,33),(8,43,15,36),(9,44,16,37),(10,45,17,38),(11,46,18,39),(12,47,19,40),(13,48,20,41),(14,49,21,42),(57,85,78,106),(58,86,79,107),(59,87,80,108),(60,88,81,109),(61,89,82,110),(62,90,83,111),(63,91,84,112),(64,92,71,99),(65,93,72,100),(66,94,73,101),(67,95,74,102),(68,96,75,103),(69,97,76,104),(70,98,77,105)], [(1,90,20,104),(2,91,21,105),(3,85,15,99),(4,86,16,100),(5,87,17,101),(6,88,18,102),(7,89,19,103),(8,92,22,106),(9,93,23,107),(10,94,24,108),(11,95,25,109),(12,96,26,110),(13,97,27,111),(14,98,28,112),(29,57,43,71),(30,58,44,72),(31,59,45,73),(32,60,46,74),(33,61,47,75),(34,62,48,76),(35,63,49,77),(36,64,50,78),(37,65,51,79),(38,66,52,80),(39,67,53,81),(40,68,54,82),(41,69,55,83),(42,70,56,84)], [(1,76),(2,77),(3,71),(4,72),(5,73),(6,74),(7,75),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D4×D7 |
| kernel | D7×C4⋊1D4 | D7×C42 | C28⋊4D4 | C28⋊D4 | C7×C4⋊1D4 | C2×D4×D7 | C4×D7 | C4⋊1D4 | C42 | C2×D4 | C4 |
| # reps | 1 | 1 | 1 | 6 | 1 | 6 | 12 | 3 | 3 | 18 | 18 |
Matrix representation of D7×C4⋊1D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 0 | 0 | 9 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,9,0,0,0,0,1,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,9,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,28,0,0,0,0,2,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,2,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D7×C4⋊1D4 in GAP, Magma, Sage, TeX
D_7\times C_4\rtimes_1D_4
% in TeX
G:=Group("D7xC4:1D4"); // GroupNames label
G:=SmallGroup(448,1167);
// by ID
G=gap.SmallGroup(448,1167);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations