direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.10D4, 2- 1+4.C14, C4≀C2⋊4C14, C4⋊Q8⋊1C14, D4.10(C7×D4), (C7×D4).44D4, (C2×C28).25D4, C4.30(D4×C14), (C7×Q8).44D4, Q8.10(C7×D4), C28.391(C2×D4), C8.C22⋊2C14, C4.10D4⋊2C14, C42.13(C2×C14), C22.17(D4×C14), C14.103C22≀C2, (C2×C28).612C23, (C4×C28).255C22, M4(2).2(C2×C14), (Q8×C14).158C22, (C7×2- 1+4).2C2, (C7×M4(2)).29C22, (C7×C4≀C2)⋊12C2, (C7×C4⋊Q8)⋊22C2, (C2×C4).6(C7×D4), C4○D4.4(C2×C14), (C7×C8.C22)⋊9C2, (C2×Q8).4(C2×C14), C2.17(C7×C22≀C2), (C7×C4.10D4)⋊8C2, (C2×C14).412(C2×D4), (C2×C4).7(C22×C14), (C7×C4○D4).34C22, SmallGroup(448,864)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.10D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >
Subgroups: 242 in 142 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, C14, C14, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, D4.10D4, C4×C28, C7×C4⋊C4, C7×M4(2), C7×SD16, C7×Q16, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4.10D4, C7×C4≀C2, C7×C4⋊Q8, C7×C8.C22, C7×2- 1+4, C7×D4.10D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D4.10D4, D4×C14, C7×C22≀C2, C7×D4.10D4
►On 112 points
Generators in S112
(1 87 58 111 103 26 33)(2 88 59 112 104 27 34)(3 81 60 105 97 28 35)(4 82 61 106 98 29 36)(5 83 62 107 99 30 37)(6 84 63 108 100 31 38)(7 85 64 109 101 32 39)(8 86 57 110 102 25 40)(9 41 77 17 49 94 69)(10 42 78 18 50 95 70)(11 43 79 19 51 96 71)(12 44 80 20 52 89 72)(13 45 73 21 53 90 65)(14 46 74 22 54 91 66)(15 47 75 23 55 92 67)(16 48 76 24 56 93 68)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(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 16)(10 11)(12 13)(14 15)(17 24)(18 19)(20 21)(22 23)(25 32)(26 27)(28 29)(30 31)(33 34)(35 36)(37 38)(39 40)(41 48)(42 43)(44 45)(46 47)(49 56)(50 51)(52 53)(54 55)(57 64)(58 59)(60 61)(62 63)(65 72)(66 67)(68 69)(70 71)(73 80)(74 75)(76 77)(78 79)(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 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 42 5 46)(2 45 6 41)(3 48 7 44)(4 43 8 47)(9 34 13 38)(10 37 14 33)(11 40 15 36)(12 35 16 39)(17 59 21 63)(18 62 22 58)(19 57 23 61)(20 60 24 64)(25 67 29 71)(26 70 30 66)(27 65 31 69)(28 68 32 72)(49 112 53 108)(50 107 54 111)(51 110 55 106)(52 105 56 109)(73 84 77 88)(74 87 78 83)(75 82 79 86)(76 85 80 81)(89 97 93 101)(90 100 94 104)(91 103 95 99)(92 98 96 102)
G:=sub<Sym(112)| (1,87,58,111,103,26,33)(2,88,59,112,104,27,34)(3,81,60,105,97,28,35)(4,82,61,106,98,29,36)(5,83,62,107,99,30,37)(6,84,63,108,100,31,38)(7,85,64,109,101,32,39)(8,86,57,110,102,25,40)(9,41,77,17,49,94,69)(10,42,78,18,50,95,70)(11,43,79,19,51,96,71)(12,44,80,20,52,89,72)(13,45,73,21,53,90,65)(14,46,74,22,54,91,66)(15,47,75,23,55,92,67)(16,48,76,24,56,93,68), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(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)(9,16)(10,11)(12,13)(14,15)(17,24)(18,19)(20,21)(22,23)(25,32)(26,27)(28,29)(30,31)(33,34)(35,36)(37,38)(39,40)(41,48)(42,43)(44,45)(46,47)(49,56)(50,51)(52,53)(54,55)(57,64)(58,59)(60,61)(62,63)(65,72)(66,67)(68,69)(70,71)(73,80)(74,75)(76,77)(78,79)(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), (1,2,3,4,5,6,7,8)(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), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,34,13,38)(10,37,14,33)(11,40,15,36)(12,35,16,39)(17,59,21,63)(18,62,22,58)(19,57,23,61)(20,60,24,64)(25,67,29,71)(26,70,30,66)(27,65,31,69)(28,68,32,72)(49,112,53,108)(50,107,54,111)(51,110,55,106)(52,105,56,109)(73,84,77,88)(74,87,78,83)(75,82,79,86)(76,85,80,81)(89,97,93,101)(90,100,94,104)(91,103,95,99)(92,98,96,102)>;
G:=Group( (1,87,58,111,103,26,33)(2,88,59,112,104,27,34)(3,81,60,105,97,28,35)(4,82,61,106,98,29,36)(5,83,62,107,99,30,37)(6,84,63,108,100,31,38)(7,85,64,109,101,32,39)(8,86,57,110,102,25,40)(9,41,77,17,49,94,69)(10,42,78,18,50,95,70)(11,43,79,19,51,96,71)(12,44,80,20,52,89,72)(13,45,73,21,53,90,65)(14,46,74,22,54,91,66)(15,47,75,23,55,92,67)(16,48,76,24,56,93,68), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(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)(9,16)(10,11)(12,13)(14,15)(17,24)(18,19)(20,21)(22,23)(25,32)(26,27)(28,29)(30,31)(33,34)(35,36)(37,38)(39,40)(41,48)(42,43)(44,45)(46,47)(49,56)(50,51)(52,53)(54,55)(57,64)(58,59)(60,61)(62,63)(65,72)(66,67)(68,69)(70,71)(73,80)(74,75)(76,77)(78,79)(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), (1,2,3,4,5,6,7,8)(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), (1,42,5,46)(2,45,6,41)(3,48,7,44)(4,43,8,47)(9,34,13,38)(10,37,14,33)(11,40,15,36)(12,35,16,39)(17,59,21,63)(18,62,22,58)(19,57,23,61)(20,60,24,64)(25,67,29,71)(26,70,30,66)(27,65,31,69)(28,68,32,72)(49,112,53,108)(50,107,54,111)(51,110,55,106)(52,105,56,109)(73,84,77,88)(74,87,78,83)(75,82,79,86)(76,85,80,81)(89,97,93,101)(90,100,94,104)(91,103,95,99)(92,98,96,102) );
G=PermutationGroup([[(1,87,58,111,103,26,33),(2,88,59,112,104,27,34),(3,81,60,105,97,28,35),(4,82,61,106,98,29,36),(5,83,62,107,99,30,37),(6,84,63,108,100,31,38),(7,85,64,109,101,32,39),(8,86,57,110,102,25,40),(9,41,77,17,49,94,69),(10,42,78,18,50,95,70),(11,43,79,19,51,96,71),(12,44,80,20,52,89,72),(13,45,73,21,53,90,65),(14,46,74,22,54,91,66),(15,47,75,23,55,92,67),(16,48,76,24,56,93,68)], [(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(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),(9,16),(10,11),(12,13),(14,15),(17,24),(18,19),(20,21),(22,23),(25,32),(26,27),(28,29),(30,31),(33,34),(35,36),(37,38),(39,40),(41,48),(42,43),(44,45),(46,47),(49,56),(50,51),(52,53),(54,55),(57,64),(58,59),(60,61),(62,63),(65,72),(66,67),(68,69),(70,71),(73,80),(74,75),(76,77),(78,79),(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)], [(1,2,3,4,5,6,7,8),(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)], [(1,42,5,46),(2,45,6,41),(3,48,7,44),(4,43,8,47),(9,34,13,38),(10,37,14,33),(11,40,15,36),(12,35,16,39),(17,59,21,63),(18,62,22,58),(19,57,23,61),(20,60,24,64),(25,67,29,71),(26,70,30,66),(27,65,31,69),(28,68,32,72),(49,112,53,108),(50,107,54,111),(51,110,55,106),(52,105,56,109),(73,84,77,88),(74,87,78,83),(75,82,79,86),(76,85,80,81),(89,97,93,101),(90,100,94,104),(91,103,95,99),(92,98,96,102)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AV | 28AW | ··· | 28BB | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 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 | D4.10D4 | C7×D4.10D4 |
| kernel | C7×D4.10D4 | C7×C4.10D4 | C7×C4≀C2 | C7×C4⋊Q8 | C7×C8.C22 | C7×2- 1+4 | D4.10D4 | C4.10D4 | C4≀C2 | C4⋊Q8 | 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×D4.10D4 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 9 | 103 | 112 | 111 |
| 57 | 66 | 1 | 1 |
| 112 | 89 | 14 | 63 |
| 101 | 51 | 64 | 78 |
| 74 | 88 | 0 | 0 |
| 31 | 82 | 82 | 63 |
| 12 | 62 | 49 | 35 |
| 112 | 89 | 14 | 63 |
| 74 | 88 | 0 | 0 |
| 91 | 53 | 94 | 12 |
| 88 | 39 | 0 | 0 |
| 39 | 25 | 0 | 0 |
| 1 | 24 | 99 | 50 |
| 25 | 64 | 39 | 14 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,9,57,1,0,103,66,0,0,112,1,0,0,111,1],[112,101,74,31,89,51,88,82,14,64,0,82,63,78,0,63],[12,112,74,91,62,89,88,53,49,14,0,94,35,63,0,12],[88,39,1,25,39,25,24,64,0,0,99,39,0,0,50,14] >;
C7×D4.10D4 in GAP, Magma, Sage, TeX
C_7\times D_4._{10}D_4 % in TeX
G:=Group("C7xD4.10D4"); // GroupNames label
G:=SmallGroup(448,864);
// by ID
G=gap.SmallGroup(448,864);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,1192,9804,4911,2468,172,7068]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=1,d^4=e^2=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^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations