direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.8D4, 2- 1+4⋊1C14, C4≀C2⋊2C14, D4.8(C7×D4), Q8.8(C7×D4), C8⋊C22⋊2C14, (C7×D4).42D4, (C2×C28).24D4, C4.28(D4×C14), (C7×Q8).42D4, C4.4D4⋊1C14, C28.389(C2×D4), C4.10D4⋊1C14, C42.12(C2×C14), C22.15(D4×C14), C14.101C22≀C2, (C2×C28).610C23, (C4×C28).254C22, M4(2).1(C2×C14), (C7×2- 1+4)⋊6C2, (D4×C14).181C22, (Q8×C14).157C22, (C7×M4(2)).28C22, (C7×C4≀C2)⋊10C2, (C2×C4).5(C7×D4), (C7×C8⋊C22)⋊9C2, C4○D4.2(C2×C14), (C2×D4).6(C2×C14), (C2×Q8).3(C2×C14), C2.15(C7×C22≀C2), (C7×C4.4D4)⋊21C2, (C7×C4.10D4)⋊7C2, (C2×C14).410(C2×D4), (C2×C4).5(C22×C14), (C7×C4○D4).32C22, SmallGroup(448,862)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.8D4
G = < a,b,c,d,e | a7=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >
Subgroups: 274 in 146 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, D4.8D4, C4×C28, C7×C22⋊C4, C7×M4(2), C7×D8, C7×SD16, D4×C14, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4.10D4, C7×C4≀C2, C7×C4.4D4, C7×C8⋊C22, C7×2- 1+4, C7×D4.8D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D4.8D4, D4×C14, C7×C22≀C2, C7×D4.8D4
►On 112 points
Generators in S112
(1 87 62 111 99 27 33)(2 88 63 112 100 28 34)(3 81 64 105 101 29 35)(4 82 57 106 102 30 36)(5 83 58 107 103 31 37)(6 84 59 108 104 32 38)(7 85 60 109 97 25 39)(8 86 61 110 98 26 40)(9 48 76 17 56 93 68)(10 41 77 18 49 94 69)(11 42 78 19 50 95 70)(12 43 79 20 51 96 71)(13 44 80 21 52 89 72)(14 45 73 22 53 90 65)(15 46 74 23 54 91 66)(16 47 75 24 55 92 67)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 87 85 83)(82 84 86 88)(89 95 93 91)(90 92 94 96)(97 103 101 99)(98 100 102 104)(105 111 109 107)(106 108 110 112)
(1 2)(3 4)(5 6)(7 8)(9 14)(10 13)(11 16)(12 15)(17 22)(18 21)(19 24)(20 23)(25 26)(27 28)(29 30)(31 32)(33 34)(35 36)(37 38)(39 40)(41 44)(42 47)(43 46)(45 48)(49 52)(50 55)(51 54)(53 56)(57 64)(58 59)(60 61)(62 63)(65 68)(66 71)(67 70)(69 72)(73 76)(74 79)(75 78)(77 80)(81 82)(83 84)(85 86)(87 88)(89 94)(90 93)(91 96)(92 95)(97 98)(99 100)(101 102)(103 104)(105 106)(107 108)(109 110)(111 112)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 48)(8 47)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 60)(18 59)(19 58)(20 57)(21 64)(22 63)(23 62)(24 61)(25 68)(26 67)(27 66)(28 65)(29 72)(30 71)(31 70)(32 69)(49 108)(50 107)(51 106)(52 105)(53 112)(54 111)(55 110)(56 109)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(89 101)(90 100)(91 99)(92 98)(93 97)(94 104)(95 103)(96 102)
G:=sub<Sym(112)| (1,87,62,111,99,27,33)(2,88,63,112,100,28,34)(3,81,64,105,101,29,35)(4,82,57,106,102,30,36)(5,83,58,107,103,31,37)(6,84,59,108,104,32,38)(7,85,60,109,97,25,39)(8,86,61,110,98,26,40)(9,48,76,17,56,93,68)(10,41,77,18,49,94,69)(11,42,78,19,50,95,70)(12,43,79,20,51,96,71)(13,44,80,21,52,89,72)(14,45,73,22,53,90,65)(15,46,74,23,54,91,66)(16,47,75,24,55,92,67), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,87,85,83)(82,84,86,88)(89,95,93,91)(90,92,94,96)(97,103,101,99)(98,100,102,104)(105,111,109,107)(106,108,110,112), (1,2)(3,4)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,44)(42,47)(43,46)(45,48)(49,52)(50,55)(51,54)(53,56)(57,64)(58,59)(60,61)(62,63)(65,68)(66,71)(67,70)(69,72)(73,76)(74,79)(75,78)(77,80)(81,82)(83,84)(85,86)(87,88)(89,94)(90,93)(91,96)(92,95)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112), (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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,48)(8,47)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,60)(18,59)(19,58)(20,57)(21,64)(22,63)(23,62)(24,61)(25,68)(26,67)(27,66)(28,65)(29,72)(30,71)(31,70)(32,69)(49,108)(50,107)(51,106)(52,105)(53,112)(54,111)(55,110)(56,109)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(89,101)(90,100)(91,99)(92,98)(93,97)(94,104)(95,103)(96,102)>;
G:=Group( (1,87,62,111,99,27,33)(2,88,63,112,100,28,34)(3,81,64,105,101,29,35)(4,82,57,106,102,30,36)(5,83,58,107,103,31,37)(6,84,59,108,104,32,38)(7,85,60,109,97,25,39)(8,86,61,110,98,26,40)(9,48,76,17,56,93,68)(10,41,77,18,49,94,69)(11,42,78,19,50,95,70)(12,43,79,20,51,96,71)(13,44,80,21,52,89,72)(14,45,73,22,53,90,65)(15,46,74,23,54,91,66)(16,47,75,24,55,92,67), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,87,85,83)(82,84,86,88)(89,95,93,91)(90,92,94,96)(97,103,101,99)(98,100,102,104)(105,111,109,107)(106,108,110,112), (1,2)(3,4)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,44)(42,47)(43,46)(45,48)(49,52)(50,55)(51,54)(53,56)(57,64)(58,59)(60,61)(62,63)(65,68)(66,71)(67,70)(69,72)(73,76)(74,79)(75,78)(77,80)(81,82)(83,84)(85,86)(87,88)(89,94)(90,93)(91,96)(92,95)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112), (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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,48)(8,47)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,40)(17,60)(18,59)(19,58)(20,57)(21,64)(22,63)(23,62)(24,61)(25,68)(26,67)(27,66)(28,65)(29,72)(30,71)(31,70)(32,69)(49,108)(50,107)(51,106)(52,105)(53,112)(54,111)(55,110)(56,109)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(89,101)(90,100)(91,99)(92,98)(93,97)(94,104)(95,103)(96,102) );
G=PermutationGroup([[(1,87,62,111,99,27,33),(2,88,63,112,100,28,34),(3,81,64,105,101,29,35),(4,82,57,106,102,30,36),(5,83,58,107,103,31,37),(6,84,59,108,104,32,38),(7,85,60,109,97,25,39),(8,86,61,110,98,26,40),(9,48,76,17,56,93,68),(10,41,77,18,49,94,69),(11,42,78,19,50,95,70),(12,43,79,20,51,96,71),(13,44,80,21,52,89,72),(14,45,73,22,53,90,65),(15,46,74,23,54,91,66),(16,47,75,24,55,92,67)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,39,37,35),(34,36,38,40),(41,43,45,47),(42,48,46,44),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,67,69,71),(66,72,70,68),(73,75,77,79),(74,80,78,76),(81,87,85,83),(82,84,86,88),(89,95,93,91),(90,92,94,96),(97,103,101,99),(98,100,102,104),(105,111,109,107),(106,108,110,112)], [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,16),(12,15),(17,22),(18,21),(19,24),(20,23),(25,26),(27,28),(29,30),(31,32),(33,34),(35,36),(37,38),(39,40),(41,44),(42,47),(43,46),(45,48),(49,52),(50,55),(51,54),(53,56),(57,64),(58,59),(60,61),(62,63),(65,68),(66,71),(67,70),(69,72),(73,76),(74,79),(75,78),(77,80),(81,82),(83,84),(85,86),(87,88),(89,94),(90,93),(91,96),(92,95),(97,98),(99,100),(101,102),(103,104),(105,106),(107,108),(109,110),(111,112)], [(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)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,48),(8,47),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,40),(17,60),(18,59),(19,58),(20,57),(21,64),(22,63),(23,62),(24,61),(25,68),(26,67),(27,66),(28,65),(29,72),(30,71),(31,70),(32,69),(49,108),(50,107),(51,106),(52,105),(53,112),(54,111),(55,110),(56,109),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(89,101),(90,100),(91,99),(92,98),(93,97),(94,104),(95,103),(96,102)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 14Y | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28AV | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D4.8D4 | C7×D4.8D4 |
| kernel | C7×D4.8D4 | C7×C4.10D4 | C7×C4≀C2 | C7×C4.4D4 | C7×C8⋊C22 | C7×2- 1+4 | D4.8D4 | C4.10D4 | C4≀C2 | C4.4D4 | C8⋊C22 | 2- 1+4 | C2×C28 | C7×D4 | C7×Q8 | C2×C4 | D4 | Q8 | 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×D4.8D4 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 15 | 0 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 15 | 98 |
| 0 | 0 | 15 | 83 |
| 0 | 0 | 0 | 98 |
| 98 | 30 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 112 |
| 15 | 83 | 0 | 0 |
| 15 | 98 | 0 | 0 |
| 1 | 111 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 83 |
| 0 | 0 | 15 | 98 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[15,15,0,0,0,98,0,0,0,0,15,15,0,0,0,98],[0,0,98,0,0,0,30,15,15,0,0,0,83,98,0,0],[0,0,15,15,0,0,83,98,1,0,0,0,111,112,0,0],[1,0,0,0,111,112,0,0,0,0,15,15,0,0,83,98] >;
C7×D4.8D4 in GAP, Magma, Sage, TeX
C_7\times D_4._8D_4
% in TeX
G:=Group("C7xD4.8D4"); // GroupNames label
G:=SmallGroup(448,862);
// by ID
G=gap.SmallGroup(448,862);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,1192,9804,4911,2468,172,7068]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=b^2*d^3>;
// generators/relations