direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4⋊C4, D14.Q8, D14.11D4, C4⋊3(C4×D7), C28⋊1(C2×C4), (C4×D7)⋊1C4, C2.3(D4×D7), C2.2(Q8×D7), C4⋊Dic7⋊11C2, Dic7⋊3(C2×C4), D14.8(C2×C4), C14.23(C2×D4), (C2×C4).30D14, C14.12(C2×Q8), Dic7⋊C4⋊11C2, C14.9(C22×C4), (C2×C28).23C22, (C2×C14).32C23, C22.16(C22×D7), (C2×Dic7).29C22, (C22×D7).34C22, C7⋊1(C2×C4⋊C4), (C7×C4⋊C4)⋊2C2, (C2×C4×D7).1C2, C2.11(C2×C4×D7), SmallGroup(224,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4⋊C4
G = < a,b,c,d | a7=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 342 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, C23, D7, C14, C4⋊C4, C4⋊C4, C22×C4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C4⋊C4, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, D7×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C4×D7, C22×D7, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4
►On 112 points
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)
(1 83 13 76)(2 84 14 77)(3 78 8 71)(4 79 9 72)(5 80 10 73)(6 81 11 74)(7 82 12 75)(15 64 22 57)(16 65 23 58)(17 66 24 59)(18 67 25 60)(19 68 26 61)(20 69 27 62)(21 70 28 63)(29 106 36 99)(30 107 37 100)(31 108 38 101)(32 109 39 102)(33 110 40 103)(34 111 41 104)(35 112 42 105)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(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)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(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)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,64,22,57)(16,65,23,58)(17,66,24,59)(18,67,25,60)(19,68,26,61)(20,69,27,62)(21,70,28,63)(29,106,36,99)(30,107,37,100)(31,108,38,101)(32,109,39,102)(33,110,40,103)(34,111,41,104)(35,112,42,105)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(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) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109)], [(1,83,13,76),(2,84,14,77),(3,78,8,71),(4,79,9,72),(5,80,10,73),(6,81,11,74),(7,82,12,75),(15,64,22,57),(16,65,23,58),(17,66,24,59),(18,67,25,60),(19,68,26,61),(20,69,27,62),(21,70,28,63),(29,106,36,99),(30,107,37,100),(31,108,38,101),(32,109,39,102),(33,110,40,103),(34,111,41,104),(35,112,42,105),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91)], [(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(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)]])
D7×C4⋊C4 is a maximal subgroup of
D4⋊(C4×D7) D14.D8 D14.SD16 Q8⋊(C4×D7) D14.1SD16 D14.Q16 C8⋊(C4×D7) D14.2SD16 D14.4SD16 C56⋊(C2×C4) D14.5D8 D14.2Q16 C14.82+ 1+4 C14.2- 1+4 C14.102+ 1+4 C42.91D14 C42.94D14 C42.95D14 C4×D4×D7 C42.108D14 D28⋊24D4 C42.113D14 C4×Q8×D7 C42.126D14 D28⋊10Q8 C42.132D14 C14.722- 1+4 C14.732- 1+4 C14.432+ 1+4 C14.172- 1+4 D28⋊22D4 C14.1182+ 1+4 C14.522+ 1+4 C14.202- 1+4 C14.212- 1+4 C14.822- 1+4 C14.832- 1+4 C14.642+ 1+4 C42.148D14 D28⋊7Q8 C42.150D14 C42.151D14 C42.152D14 C42.153D14 C42.161D14 C42.162D14 C42.163D14 D28⋊12D4 C42.174D14 D28⋊9Q8
D7×C4⋊C4 is a maximal quotient of
C14.(C4×Q8) Dic7⋊C42 C7⋊(C42⋊8C4) D14⋊(C4⋊C4) D14⋊C4⋊C4 C28⋊M4(2) C42.30D14 (C8×D7)⋊C4 C8⋊(C4×D7) C8.27(C4×D7) C56⋊(C2×C4) M4(2).25D14 Dic7⋊(C4⋊C4) (C4×Dic7)⋊8C4 C4⋊(D14⋊C4) D14⋊C4⋊6C4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| 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 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | D14 | C4×D7 | D4×D7 | Q8×D7 |
| kernel | D7×C4⋊C4 | Dic7⋊C4 | C4⋊Dic7 | C7×C4⋊C4 | C2×C4×D7 | C4×D7 | D14 | D14 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 8 | 2 | 2 | 3 | 9 | 12 | 3 | 3 |
Matrix representation of D7×C4⋊C4 ►in GL4(𝔽29) generated by
| 0 | 1 | 0 | 0 |
| 28 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
G:=sub<GL(4,GF(29))| [0,28,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,12,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,28,0,0,1,0] >;
D7×C4⋊C4 in GAP, Magma, Sage, TeX
D_7\times C_4\rtimes C_4
% in TeX
G:=Group("D7xC4:C4"); // GroupNames label
G:=SmallGroup(224,86);
// by ID
G=gap.SmallGroup(224,86);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations