direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C4⋊D4, C28⋊9D4, C4⋊2(C7×D4), C4⋊C4⋊2C14, (C2×C14)⋊4D4, (C2×D4)⋊2C14, C2.5(D4×C14), C22⋊1(C7×D4), (D4×C14)⋊11C2, C22⋊C4⋊3C14, (C22×C4)⋊4C14, C14.68(C2×D4), (C22×C28)⋊11C2, C23.8(C2×C14), C14.41(C4○D4), (C2×C14).76C23, (C2×C28).123C22, (C22×C14).27C22, C22.11(C22×C14), (C7×C4⋊C4)⋊11C2, C2.4(C7×C4○D4), (C2×C4).3(C2×C14), (C7×C22⋊C4)⋊11C2, SmallGroup(224,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊D4
G = < a,b,c,d | a7=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C14, C7×C4⋊D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C7×D4, C22×C14, D4×C14, C7×C4○D4, C7×C4⋊D4
►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 78 39 94)(2 79 40 95)(3 80 41 96)(4 81 42 97)(5 82 36 98)(6 83 37 92)(7 84 38 93)(8 62 23 77)(9 63 24 71)(10 57 25 72)(11 58 26 73)(12 59 27 74)(13 60 28 75)(14 61 22 76)(15 66 106 50)(16 67 107 51)(17 68 108 52)(18 69 109 53)(19 70 110 54)(20 64 111 55)(21 65 112 56)(29 100 44 85)(30 101 45 86)(31 102 46 87)(32 103 47 88)(33 104 48 89)(34 105 49 90)(35 99 43 91)
(1 66 35 63)(2 67 29 57)(3 68 30 58)(4 69 31 59)(5 70 32 60)(6 64 33 61)(7 65 34 62)(8 84 21 105)(9 78 15 99)(10 79 16 100)(11 80 17 101)(12 81 18 102)(13 82 19 103)(14 83 20 104)(22 92 111 89)(23 93 112 90)(24 94 106 91)(25 95 107 85)(26 96 108 86)(27 97 109 87)(28 98 110 88)(36 54 47 75)(37 55 48 76)(38 56 49 77)(39 50 43 71)(40 51 44 72)(41 52 45 73)(42 53 46 74)
(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
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), (1,78,39,94)(2,79,40,95)(3,80,41,96)(4,81,42,97)(5,82,36,98)(6,83,37,92)(7,84,38,93)(8,62,23,77)(9,63,24,71)(10,57,25,72)(11,58,26,73)(12,59,27,74)(13,60,28,75)(14,61,22,76)(15,66,106,50)(16,67,107,51)(17,68,108,52)(18,69,109,53)(19,70,110,54)(20,64,111,55)(21,65,112,56)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,99,43,91), (1,66,35,63)(2,67,29,57)(3,68,30,58)(4,69,31,59)(5,70,32,60)(6,64,33,61)(7,65,34,62)(8,84,21,105)(9,78,15,99)(10,79,16,100)(11,80,17,101)(12,81,18,102)(13,82,19,103)(14,83,20,104)(22,92,111,89)(23,93,112,90)(24,94,106,91)(25,95,107,85)(26,96,108,86)(27,97,109,87)(28,98,110,88)(36,54,47,75)(37,55,48,76)(38,56,49,77)(39,50,43,71)(40,51,44,72)(41,52,45,73)(42,53,46,74), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)>;
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), (1,78,39,94)(2,79,40,95)(3,80,41,96)(4,81,42,97)(5,82,36,98)(6,83,37,92)(7,84,38,93)(8,62,23,77)(9,63,24,71)(10,57,25,72)(11,58,26,73)(12,59,27,74)(13,60,28,75)(14,61,22,76)(15,66,106,50)(16,67,107,51)(17,68,108,52)(18,69,109,53)(19,70,110,54)(20,64,111,55)(21,65,112,56)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,99,43,91), (1,66,35,63)(2,67,29,57)(3,68,30,58)(4,69,31,59)(5,70,32,60)(6,64,33,61)(7,65,34,62)(8,84,21,105)(9,78,15,99)(10,79,16,100)(11,80,17,101)(12,81,18,102)(13,82,19,103)(14,83,20,104)(22,92,111,89)(23,93,112,90)(24,94,106,91)(25,95,107,85)(26,96,108,86)(27,97,109,87)(28,98,110,88)(36,54,47,75)(37,55,48,76)(38,56,49,77)(39,50,43,71)(40,51,44,72)(41,52,45,73)(42,53,46,74), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99) );
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)], [(1,78,39,94),(2,79,40,95),(3,80,41,96),(4,81,42,97),(5,82,36,98),(6,83,37,92),(7,84,38,93),(8,62,23,77),(9,63,24,71),(10,57,25,72),(11,58,26,73),(12,59,27,74),(13,60,28,75),(14,61,22,76),(15,66,106,50),(16,67,107,51),(17,68,108,52),(18,69,109,53),(19,70,110,54),(20,64,111,55),(21,65,112,56),(29,100,44,85),(30,101,45,86),(31,102,46,87),(32,103,47,88),(33,104,48,89),(34,105,49,90),(35,99,43,91)], [(1,66,35,63),(2,67,29,57),(3,68,30,58),(4,69,31,59),(5,70,32,60),(6,64,33,61),(7,65,34,62),(8,84,21,105),(9,78,15,99),(10,79,16,100),(11,80,17,101),(12,81,18,102),(13,82,19,103),(14,83,20,104),(22,92,111,89),(23,93,112,90),(24,94,106,91),(25,95,107,85),(26,96,108,86),(27,97,109,87),(28,98,110,88),(36,54,47,75),(37,55,48,76),(38,56,49,77),(39,50,43,71),(40,51,44,72),(41,52,45,73),(42,53,46,74)], [(8,112),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)]])
C7×C4⋊D4 is a maximal subgroup of
(D4×C14)⋊C4 (C2×C14).D8 C4⋊D4.D7 (C2×D4).D14 D28⋊16D4 D28⋊17D4 C7⋊C8⋊22D4 C4⋊D4⋊D7 Dic14⋊17D4 C7⋊C8⋊23D4 C7⋊C8⋊5D4 C28⋊(C4○D4) C14.682- 1+4 Dic14⋊19D4 Dic14⋊20D4 C4⋊C4.178D14 C14.342+ 1+4 C14.352+ 1+4 C14.712- 1+4 C14.372+ 1+4 C4⋊C4⋊21D14 C14.382+ 1+4 C14.722- 1+4 D28⋊19D4 C14.402+ 1+4 C14.732- 1+4 D28⋊20D4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C14.452+ 1+4 C14.462+ 1+4 C14.1152+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C14.492+ 1+4 C7×D42
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C4⋊D4 | C7×C22⋊C4 | C7×C4⋊C4 | C22×C28 | D4×C14 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 6 | 12 | 6 | 6 | 18 | 2 | 2 | 2 | 12 | 12 | 12 |
Matrix representation of C7×C4⋊D4 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 17 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 23 | 27 |
| 0 | 0 | 4 | 6 |
| 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 23 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 23 | 28 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,25,0,0,0,0,25],[0,17,0,0,17,0,0,0,0,0,23,4,0,0,27,6],[0,1,0,0,28,0,0,0,0,0,1,23,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,1,23,0,0,0,28] >;
C7×C4⋊D4 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes D_4
% in TeX
G:=Group("C7xC4:D4"); // GroupNames label
G:=SmallGroup(224,156);
// by ID
G=gap.SmallGroup(224,156);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,343,2090]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations