metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊4D14, SD16⋊6D14, D56⋊3C22, C56.5C23, C28.24C24, M4(2)⋊12D14, D28.17C23, Dic14.17C23, C8⋊D14⋊3C2, D56⋊C2⋊3C2, D4⋊D7⋊7C22, (C2×Q8)⋊23D14, (C4×D7).45D4, C4.192(D4×D7), (C8×D7)⋊5C22, C7⋊C8.12C23, Q8⋊D7⋊6C22, C8.C22⋊6D7, Q8.D14⋊1C2, C8.5(C22×D7), Q16⋊D7⋊2C2, C4○D4.30D14, D14.55(C2×D4), C28.245(C2×D4), (D4×D7)⋊10C22, C56⋊C2⋊6C22, C8⋊D7⋊6C22, (D7×M4(2))⋊4C2, D4⋊D14⋊10C2, (C7×Q16)⋊2C22, (Q8×D7)⋊12C22, C7⋊Q16⋊5C22, C22.49(D4×D7), C4.24(C23×D7), SD16⋊3D7⋊3C2, (C2×D28)⋊37C22, C7⋊4(D8⋊C22), Dic7.62(C2×D4), (Q8×C14)⋊21C22, (C7×SD16)⋊6C22, (C22×D7).44D4, (C7×D4).17C23, (C4×D7).31C23, D4.17(C22×D7), (C7×Q8).17C23, Q8.17(C22×D7), C28.C23⋊10C2, D4⋊2D7⋊11C22, (C2×C28).115C23, (C2×Dic7).196D4, Q8⋊2D7⋊11C22, C4○D28.31C22, C14.125(C22×D4), (C7×M4(2))⋊6C22, C4.Dic7⋊15C22, C2.98(C2×D4×D7), (D7×C4○D4)⋊5C2, (C2×C14).70(C2×D4), (C7×C8.C22)⋊2C2, (C2×Q8⋊2D7)⋊17C2, (C2×C4×D7).163C22, (C2×C4).99(C22×D7), (C7×C4○D4).26C22, SmallGroup(448,1230)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊C22
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a-1, cac=a29, dad=a13, cbc=a28b, dbd=a40b, cd=dc >
Subgroups: 1356 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C8.C22, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, C22×D7, D8⋊C22, C8×D7, C8⋊D7, C56⋊C2, D56, C4.Dic7, D4⋊D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, Q8⋊2D7, Q8⋊2D7, Q8×C14, C7×C4○D4, D7×M4(2), C8⋊D14, D56⋊C2, SD16⋊3D7, Q16⋊D7, Q8.D14, C28.C23, D4⋊D14, C7×C8.C22, C2×Q8⋊2D7, D7×C4○D4, D56⋊C22
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D8⋊C22, D4×D7, C23×D7, C2×D4×D7, D56⋊C22
►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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
(1 85)(2 98)(3 111)(4 68)(5 81)(6 94)(7 107)(8 64)(9 77)(10 90)(11 103)(12 60)(13 73)(14 86)(15 99)(16 112)(17 69)(18 82)(19 95)(20 108)(21 65)(22 78)(23 91)(24 104)(25 61)(26 74)(27 87)(28 100)(29 57)(30 70)(31 83)(32 96)(33 109)(34 66)(35 79)(36 92)(37 105)(38 62)(39 75)(40 88)(41 101)(42 58)(43 71)(44 84)(45 97)(46 110)(47 67)(48 80)(49 93)(50 106)(51 63)(52 76)(53 89)(54 102)(55 59)(56 72)
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,85)(2,98)(3,111)(4,68)(5,81)(6,94)(7,107)(8,64)(9,77)(10,90)(11,103)(12,60)(13,73)(14,86)(15,99)(16,112)(17,69)(18,82)(19,95)(20,108)(21,65)(22,78)(23,91)(24,104)(25,61)(26,74)(27,87)(28,100)(29,57)(30,70)(31,83)(32,96)(33,109)(34,66)(35,79)(36,92)(37,105)(38,62)(39,75)(40,88)(41,101)(42,58)(43,71)(44,84)(45,97)(46,110)(47,67)(48,80)(49,93)(50,106)(51,63)(52,76)(53,89)(54,102)(55,59)(56,72)>;
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,85)(2,98)(3,111)(4,68)(5,81)(6,94)(7,107)(8,64)(9,77)(10,90)(11,103)(12,60)(13,73)(14,86)(15,99)(16,112)(17,69)(18,82)(19,95)(20,108)(21,65)(22,78)(23,91)(24,104)(25,61)(26,74)(27,87)(28,100)(29,57)(30,70)(31,83)(32,96)(33,109)(34,66)(35,79)(36,92)(37,105)(38,62)(39,75)(40,88)(41,101)(42,58)(43,71)(44,84)(45,97)(46,110)(47,67)(48,80)(49,93)(50,106)(51,63)(52,76)(53,89)(54,102)(55,59)(56,72) );
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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99)], [(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)], [(1,85),(2,98),(3,111),(4,68),(5,81),(6,94),(7,107),(8,64),(9,77),(10,90),(11,103),(12,60),(13,73),(14,86),(15,99),(16,112),(17,69),(18,82),(19,95),(20,108),(21,65),(22,78),(23,91),(24,104),(25,61),(26,74),(27,87),(28,100),(29,57),(30,70),(31,83),(32,96),(33,109),(34,66),(35,79),(36,92),(37,105),(38,62),(39,75),(40,88),(41,101),(42,58),(43,71),(44,84),(45,97),(46,110),(47,67),(48,80),(49,93),(50,106),(51,63),(52,76),(53,89),(54,102),(55,59),(56,72)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D8⋊C22 | D4×D7 | D4×D7 | D56⋊C22 |
| kernel | D56⋊C22 | D7×M4(2) | C8⋊D14 | D56⋊C2 | SD16⋊3D7 | Q16⋊D7 | Q8.D14 | C28.C23 | D4⋊D14 | C7×C8.C22 | C2×Q8⋊2D7 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of D56⋊C22 ►in GL8(𝔽113)
| 36 | 59 | 110 | 65 | 0 | 0 | 0 | 0 |
| 22 | 49 | 27 | 23 | 0 | 0 | 0 | 0 |
| 72 | 29 | 65 | 54 | 0 | 0 | 0 | 0 |
| 62 | 61 | 41 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 106 | 9 | 83 |
| 0 | 0 | 0 | 0 | 0 | 100 | 0 | 25 |
| 0 | 0 | 0 | 0 | 0 | 87 | 98 | 50 |
| 0 | 0 | 0 | 0 | 15 | 61 | 0 | 13 |
| 98 | 95 | 110 | 10 | 0 | 0 | 0 | 0 |
| 15 | 34 | 27 | 93 | 0 | 0 | 0 | 0 |
| 50 | 55 | 65 | 16 | 0 | 0 | 0 | 0 |
| 110 | 62 | 41 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 106 | 9 | 83 |
| 0 | 0 | 0 | 0 | 88 | 100 | 98 | 25 |
| 0 | 0 | 0 | 0 | 0 | 4 | 98 | 50 |
| 0 | 0 | 0 | 0 | 2 | 61 | 60 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 105 | 0 | 112 |
| 14 | 75 | 22 | 58 | 0 | 0 | 0 | 0 |
| 82 | 51 | 91 | 48 | 0 | 0 | 0 | 0 |
| 95 | 101 | 19 | 62 | 0 | 0 | 0 | 0 |
| 102 | 90 | 51 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 109 | 15 | 63 |
| 0 | 0 | 0 | 0 | 2 | 0 | 9 | 98 |
G:=sub<GL(8,GF(113))| [36,22,72,62,0,0,0,0,59,49,29,61,0,0,0,0,110,27,65,41,0,0,0,0,65,23,54,76,0,0,0,0,0,0,0,0,15,0,0,15,0,0,0,0,106,100,87,61,0,0,0,0,9,0,98,0,0,0,0,0,83,25,50,13],[98,15,50,110,0,0,0,0,95,34,55,62,0,0,0,0,110,27,65,41,0,0,0,0,10,93,16,29,0,0,0,0,0,0,0,0,15,88,0,2,0,0,0,0,106,100,4,61,0,0,0,0,9,98,98,60,0,0,0,0,83,25,50,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,105,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[14,82,95,102,0,0,0,0,75,51,101,90,0,0,0,0,22,91,19,51,0,0,0,0,58,48,62,29,0,0,0,0,0,0,0,0,15,88,0,2,0,0,0,0,18,98,109,0,0,0,0,0,0,0,15,9,0,0,0,0,0,0,63,98] >;
D56⋊C22 in GAP, Magma, Sage, TeX
D_{56}\rtimes C_2^2 % in TeX
G:=Group("D56:C2^2"); // GroupNames label
G:=SmallGroup(448,1230);
// by ID
G=gap.SmallGroup(448,1230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^56=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^29,d*a*d=a^13,c*b*c=a^28*b,d*b*d=a^40*b,c*d=d*c>;
// generators/relations