direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C42, C28⋊5(C2×C4), (C4×C28)⋊8C2, C7⋊1(C2×C42), Dic7⋊5(C2×C4), D14.7(C2×C4), (C2×C4).95D14, (C4×Dic7)⋊17C2, C14.2(C22×C4), (C2×C14).12C23, C22.9(C22×D7), (C2×C28).109C22, (C2×Dic7).45C22, (C22×D7).32C22, C2.1(C2×C4×D7), (C2×C4×D7).11C2, SmallGroup(224,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C42 |
Generators and relations for D7×C42
G = < a,b,c,d | a4=b4=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 342 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, D7, C14, C42, C42, C22×C4, Dic7, C28, D14, C2×C14, C2×C42, C4×D7, C2×Dic7, C2×C28, C22×D7, C4×Dic7, C4×C28, C2×C4×D7, D7×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, D14, C2×C42, C4×D7, C22×D7, C2×C4×D7, D7×C42
►On 112 points
Generators in S112
(1 69 13 62)(2 70 14 63)(3 64 8 57)(4 65 9 58)(5 66 10 59)(6 67 11 60)(7 68 12 61)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 55 20 41)(2 56 21 42)(3 50 15 36)(4 51 16 37)(5 52 17 38)(6 53 18 39)(7 54 19 40)(8 43 22 29)(9 44 23 30)(10 45 24 31)(11 46 25 32)(12 47 26 33)(13 48 27 34)(14 49 28 35)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)(64 73)(65 72)(66 71)(67 77)(68 76)(69 75)(70 74)(85 108)(86 107)(87 106)(88 112)(89 111)(90 110)(91 109)(92 101)(93 100)(94 99)(95 105)(96 104)(97 103)(98 102)
G:=sub<Sym(112)| (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,55,20,41)(2,56,21,42)(3,50,15,36)(4,51,16,37)(5,52,17,38)(6,53,18,39)(7,54,19,40)(8,43,22,29)(9,44,23,30)(10,45,24,31)(11,46,25,32)(12,47,26,33)(13,48,27,34)(14,49,28,35)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,80)(58,79)(59,78)(60,84)(61,83)(62,82)(63,81)(64,73)(65,72)(66,71)(67,77)(68,76)(69,75)(70,74)(85,108)(86,107)(87,106)(88,112)(89,111)(90,110)(91,109)(92,101)(93,100)(94,99)(95,105)(96,104)(97,103)(98,102)>;
G:=Group( (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,55,20,41)(2,56,21,42)(3,50,15,36)(4,51,16,37)(5,52,17,38)(6,53,18,39)(7,54,19,40)(8,43,22,29)(9,44,23,30)(10,45,24,31)(11,46,25,32)(12,47,26,33)(13,48,27,34)(14,49,28,35)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,80)(58,79)(59,78)(60,84)(61,83)(62,82)(63,81)(64,73)(65,72)(66,71)(67,77)(68,76)(69,75)(70,74)(85,108)(86,107)(87,106)(88,112)(89,111)(90,110)(91,109)(92,101)(93,100)(94,99)(95,105)(96,104)(97,103)(98,102) );
G=PermutationGroup([[(1,69,13,62),(2,70,14,63),(3,64,8,57),(4,65,9,58),(5,66,10,59),(6,67,11,60),(7,68,12,61),(15,78,22,71),(16,79,23,72),(17,80,24,73),(18,81,25,74),(19,82,26,75),(20,83,27,76),(21,84,28,77),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105)], [(1,55,20,41),(2,56,21,42),(3,50,15,36),(4,51,16,37),(5,52,17,38),(6,53,18,39),(7,54,19,40),(8,43,22,29),(9,44,23,30),(10,45,24,31),(11,46,25,32),(12,47,26,33),(13,48,27,34),(14,49,28,35),(57,106,71,92),(58,107,72,93),(59,108,73,94),(60,109,74,95),(61,110,75,96),(62,111,76,97),(63,112,77,98),(64,99,78,85),(65,100,79,86),(66,101,80,87),(67,102,81,88),(68,103,82,89),(69,104,83,90),(70,105,84,91)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,17),(9,16),(10,15),(11,21),(12,20),(13,19),(14,18),(29,52),(30,51),(31,50),(32,56),(33,55),(34,54),(35,53),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46),(57,80),(58,79),(59,78),(60,84),(61,83),(62,82),(63,81),(64,73),(65,72),(66,71),(67,77),(68,76),(69,75),(70,74),(85,108),(86,107),(87,106),(88,112),(89,111),(90,110),(91,109),(92,101),(93,100),(94,99),(95,105),(96,104),(97,103),(98,102)]])
D7×C42 is a maximal subgroup of
C42.282D14 C42.182D14 Dic7.C42 C42.200D14 C42.202D14 C28⋊M4(2) C42.188D14 C42.93D14 C42.228D14 C42.229D14 C42.232D14 C42.131D14 C42.233D14 C42.234D14 C42.236D14 C42.237D14 C42.189D14 C42.238D14 C42.240D14 C42.241D14
D7×C42 is a maximal quotient of
Dic7.5C42 Dic7⋊C42 D14⋊C42 D14.C42 Dic7.C42 D14.4C42
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4X | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D7 | D14 | C4×D7 |
| kernel | D7×C42 | C4×Dic7 | C4×C28 | C2×C4×D7 | C4×D7 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 3 | 9 | 36 |
Matrix representation of D7×C42 ►in GL3(𝔽29) generated by
| 1 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 17 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 28 | 18 |
| 1 | 0 | 0 |
| 0 | 0 | 28 |
| 0 | 28 | 0 |
G:=sub<GL(3,GF(29))| [1,0,0,0,17,0,0,0,17],[17,0,0,0,28,0,0,0,28],[1,0,0,0,0,28,0,1,18],[1,0,0,0,0,28,0,28,0] >;
D7×C42 in GAP, Magma, Sage, TeX
D_7\times C_4^2
% in TeX
G:=Group("D7xC4^2"); // GroupNames label
G:=SmallGroup(224,66);
// by ID
G=gap.SmallGroup(224,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations