direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.9D4, 2+ 1+4.1C14, C4≀C2⋊3C14, D4.9(C7×D4), Q8.9(C7×D4), C42⋊3(C2×C14), (C7×D4).43D4, C4.29(D4×C14), (C7×Q8).43D4, C23.6(C7×D4), (C4×C28)⋊36C22, C4.D4⋊2C14, C28.390(C2×D4), C4.4D4⋊2C14, C8.C22⋊1C14, (C22×C14).6D4, M4(2)⋊2(C2×C14), (Q8×C14)⋊27C22, C22.16(D4×C14), C14.102C22≀C2, (C2×C28).611C23, (D4×C14).182C22, (C7×M4(2))⋊18C22, (C7×2+ 1+4).3C2, (C7×C4≀C2)⋊11C2, (C2×Q8)⋊2(C2×C14), C4○D4.3(C2×C14), (C2×D4).7(C2×C14), (C7×C4.D4)⋊8C2, (C7×C8.C22)⋊8C2, C2.16(C7×C22≀C2), (C2×C14).411(C2×D4), (C7×C4.4D4)⋊22C2, (C2×C4).6(C22×C14), (C7×C4○D4).33C22, SmallGroup(448,863)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.9D4
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >
Subgroups: 306 in 152 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D4.9D4, C4×C28, C7×C22⋊C4, C7×M4(2), C7×SD16, C7×Q16, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4.D4, C7×C4≀C2, C7×C4.4D4, C7×C8.C22, C7×2+ 1+4, C7×D4.9D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D4.9D4, D4×C14, C7×C22≀C2, C7×D4.9D4
►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 47 35 39)(2 48 29 40)(3 49 30 41)(4 43 31 42)(5 44 32 36)(6 45 33 37)(7 46 34 38)(8 112 18 23)(9 106 19 24)(10 107 20 25)(11 108 21 26)(12 109 15 27)(13 110 16 28)(14 111 17 22)(50 63 75 66)(51 57 76 67)(52 58 77 68)(53 59 71 69)(54 60 72 70)(55 61 73 64)(56 62 74 65)(78 91 103 94)(79 85 104 95)(80 86 105 96)(81 87 99 97)(82 88 100 98)(83 89 101 92)(84 90 102 93)
(1 103)(2 104)(3 105)(4 99)(5 100)(6 101)(7 102)(8 56)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 61)(23 62)(24 63)(25 57)(26 58)(27 59)(28 60)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 78)(36 98)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 87)(44 88)(45 89)(46 90)(47 91)(48 85)(49 86)(64 111)(65 112)(66 106)(67 107)(68 108)(69 109)(70 110)
(8 112 18 23)(9 106 19 24)(10 107 20 25)(11 108 21 26)(12 109 15 27)(13 110 16 28)(14 111 17 22)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 94 103 91)(79 95 104 85)(80 96 105 86)(81 97 99 87)(82 98 100 88)(83 92 101 89)(84 93 102 90)
(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 18 102)(9 78 19 103)(10 79 20 104)(11 80 21 105)(12 81 15 99)(13 82 16 100)(14 83 17 101)(22 89 111 92)(23 90 112 93)(24 91 106 94)(25 85 107 95)(26 86 108 96)(27 87 109 97)(28 88 110 98)(36 54 44 72)(37 55 45 73)(38 56 46 74)(39 50 47 75)(40 51 48 76)(41 52 49 77)(42 53 43 71)
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), (1,47,35,39)(2,48,29,40)(3,49,30,41)(4,43,31,42)(5,44,32,36)(6,45,33,37)(7,46,34,38)(8,112,18,23)(9,106,19,24)(10,107,20,25)(11,108,21,26)(12,109,15,27)(13,110,16,28)(14,111,17,22)(50,63,75,66)(51,57,76,67)(52,58,77,68)(53,59,71,69)(54,60,72,70)(55,61,73,64)(56,62,74,65)(78,91,103,94)(79,85,104,95)(80,86,105,96)(81,87,99,97)(82,88,100,98)(83,89,101,92)(84,90,102,93), (1,103)(2,104)(3,105)(4,99)(5,100)(6,101)(7,102)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,61)(23,62)(24,63)(25,57)(26,58)(27,59)(28,60)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,78)(36,98)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,87)(44,88)(45,89)(46,90)(47,91)(48,85)(49,86)(64,111)(65,112)(66,106)(67,107)(68,108)(69,109)(70,110), (8,112,18,23)(9,106,19,24)(10,107,20,25)(11,108,21,26)(12,109,15,27)(13,110,16,28)(14,111,17,22)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94,103,91)(79,95,104,85)(80,96,105,86)(81,97,99,87)(82,98,100,88)(83,92,101,89)(84,93,102,90), (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)(8,84,18,102)(9,78,19,103)(10,79,20,104)(11,80,21,105)(12,81,15,99)(13,82,16,100)(14,83,17,101)(22,89,111,92)(23,90,112,93)(24,91,106,94)(25,85,107,95)(26,86,108,96)(27,87,109,97)(28,88,110,98)(36,54,44,72)(37,55,45,73)(38,56,46,74)(39,50,47,75)(40,51,48,76)(41,52,49,77)(42,53,43,71)>;
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), (1,47,35,39)(2,48,29,40)(3,49,30,41)(4,43,31,42)(5,44,32,36)(6,45,33,37)(7,46,34,38)(8,112,18,23)(9,106,19,24)(10,107,20,25)(11,108,21,26)(12,109,15,27)(13,110,16,28)(14,111,17,22)(50,63,75,66)(51,57,76,67)(52,58,77,68)(53,59,71,69)(54,60,72,70)(55,61,73,64)(56,62,74,65)(78,91,103,94)(79,85,104,95)(80,86,105,96)(81,87,99,97)(82,88,100,98)(83,89,101,92)(84,90,102,93), (1,103)(2,104)(3,105)(4,99)(5,100)(6,101)(7,102)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,61)(23,62)(24,63)(25,57)(26,58)(27,59)(28,60)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,78)(36,98)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,87)(44,88)(45,89)(46,90)(47,91)(48,85)(49,86)(64,111)(65,112)(66,106)(67,107)(68,108)(69,109)(70,110), (8,112,18,23)(9,106,19,24)(10,107,20,25)(11,108,21,26)(12,109,15,27)(13,110,16,28)(14,111,17,22)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94,103,91)(79,95,104,85)(80,96,105,86)(81,97,99,87)(82,98,100,88)(83,92,101,89)(84,93,102,90), (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)(8,84,18,102)(9,78,19,103)(10,79,20,104)(11,80,21,105)(12,81,15,99)(13,82,16,100)(14,83,17,101)(22,89,111,92)(23,90,112,93)(24,91,106,94)(25,85,107,95)(26,86,108,96)(27,87,109,97)(28,88,110,98)(36,54,44,72)(37,55,45,73)(38,56,46,74)(39,50,47,75)(40,51,48,76)(41,52,49,77)(42,53,43,71) );
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)], [(1,47,35,39),(2,48,29,40),(3,49,30,41),(4,43,31,42),(5,44,32,36),(6,45,33,37),(7,46,34,38),(8,112,18,23),(9,106,19,24),(10,107,20,25),(11,108,21,26),(12,109,15,27),(13,110,16,28),(14,111,17,22),(50,63,75,66),(51,57,76,67),(52,58,77,68),(53,59,71,69),(54,60,72,70),(55,61,73,64),(56,62,74,65),(78,91,103,94),(79,85,104,95),(80,86,105,96),(81,87,99,97),(82,88,100,98),(83,89,101,92),(84,90,102,93)], [(1,103),(2,104),(3,105),(4,99),(5,100),(6,101),(7,102),(8,56),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,61),(23,62),(24,63),(25,57),(26,58),(27,59),(28,60),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,78),(36,98),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,87),(44,88),(45,89),(46,90),(47,91),(48,85),(49,86),(64,111),(65,112),(66,106),(67,107),(68,108),(69,109),(70,110)], [(8,112,18,23),(9,106,19,24),(10,107,20,25),(11,108,21,26),(12,109,15,27),(13,110,16,28),(14,111,17,22),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,94,103,91),(79,95,104,85),(80,96,105,86),(81,97,99,87),(82,98,100,88),(83,92,101,89),(84,93,102,90)], [(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),(8,84,18,102),(9,78,19,103),(10,79,20,104),(11,80,21,105),(12,81,15,99),(13,82,16,100),(14,83,17,101),(22,89,111,92),(23,90,112,93),(24,91,106,94),(25,85,107,95),(26,86,108,96),(27,87,109,97),(28,88,110,98),(36,54,44,72),(37,55,45,73),(38,56,46,74),(39,50,47,75),(40,51,48,76),(41,52,49,77),(42,53,43,71)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14AJ | 28A | ··· | 28L | 28M | ··· | 28AJ | 28AK | ··· | 28AP | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | ··· | 1 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C7×D4 | C7×D4 | C7×D4 | D4.9D4 | C7×D4.9D4 |
| kernel | C7×D4.9D4 | C7×C4.D4 | C7×C4≀C2 | C7×C4.4D4 | C7×C8.C22 | C7×2+ 1+4 | D4.9D4 | C4.D4 | C4≀C2 | C4.4D4 | C8.C22 | 2+ 1+4 | C7×D4 | C7×Q8 | C22×C14 | D4 | Q8 | C23 | C7 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 12 | 6 | 2 | 2 | 2 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of C7×D4.9D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 15 | 0 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 98 | 15 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 98 | 15 |
| 15 | 0 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 112 | 2 | 0 | 0 |
| 112 | 1 | 0 | 0 |
| 0 | 0 | 15 | 83 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,49,0,0,0,0,49],[15,15,0,0,0,98,0,0,0,0,98,98,0,0,0,15],[0,0,15,15,0,0,0,98,98,98,0,0,0,15,0,0],[1,1,0,0,0,112,0,0,0,0,15,0,0,0,0,15],[112,112,0,0,2,1,0,0,0,0,15,0,0,0,83,98] >;
C7×D4.9D4 in GAP, Magma, Sage, TeX
C_7\times D_4._9D_4
% in TeX
G:=Group("C7xD4.9D4"); // GroupNames label
G:=SmallGroup(448,863);
// by ID
G=gap.SmallGroup(448,863);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,9804,4911,2468,172,7068]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations