metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊4D14, SD16⋊6D14, D56⋊3C22, C56.5C23, C28.24C24, M4(2)⋊12D14, D28.17C23, Dic14.17C23, D56⋊C2⋊3C2, C8⋊D14⋊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, C8⋊D7⋊6C22, C56⋊C2⋊6C22, (D7×M4(2))⋊4C2, D4⋊D14⋊10C2, (Q8×D7)⋊12C22, (C7×Q16)⋊2C22, 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, (C4×D7).31C23, (C7×D4).17C23, (C22×D7).44D4, 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
Subgroups: 1356 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×13], Q8, Q8 [×2], Q8 [×3], C23 [×3], D7 [×5], C14, C14 [×2], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×4], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×2], C22×C4 [×3], C2×D4 [×4], C2×Q8, C2×Q8, C4○D4, C4○D4 [×11], Dic7 [×2], Dic7, C28 [×2], C28 [×3], D14 [×2], D14 [×8], C2×C14, C2×C14, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22, C8.C22 [×3], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14, Dic14, C4×D7 [×4], C4×D7 [×7], D28, D28 [×2], D28 [×6], C2×Dic7, C2×Dic7, C7⋊D4 [×3], C2×C28, C2×C28 [×2], C7×D4, C7×D4, C7×Q8, C7×Q8 [×2], C7×Q8, C22×D7, C22×D7 [×2], D8⋊C22, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56 [×2], C4.Dic7, D4⋊D7 [×2], Q8⋊D7 [×4], C7⋊Q16 [×2], C7×M4(2), C7×SD16 [×2], C7×Q16 [×2], C2×C4×D7, C2×C4×D7 [×2], C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, Q8⋊2D7 [×4], Q8⋊2D7 [×2], Q8×C14, C7×C4○D4, D7×M4(2), C8⋊D14, D56⋊C2 [×2], SD16⋊3D7 [×2], Q16⋊D7 [×2], Q8.D14 [×2], C28.C23, D4⋊D14, C7×C8.C22, C2×Q8⋊2D7, D7×C4○D4, D56⋊C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D8⋊C22, D4×D7 [×2], C23×D7, C2×D4×D7, D56⋊C22
Generators and relations
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a-1, cac=a29, dad=a13, cbc=a28b, dbd=a40b, cd=dc >
►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 62)(58 61)(59 60)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)
(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)(57 85)(59 87)(61 89)(63 91)(65 93)(67 95)(69 97)(71 99)(73 101)(75 103)(77 105)(79 107)(81 109)(83 111)
(1 74)(2 87)(3 100)(4 57)(5 70)(6 83)(7 96)(8 109)(9 66)(10 79)(11 92)(12 105)(13 62)(14 75)(15 88)(16 101)(17 58)(18 71)(19 84)(20 97)(21 110)(22 67)(23 80)(24 93)(25 106)(26 63)(27 76)(28 89)(29 102)(30 59)(31 72)(32 85)(33 98)(34 111)(35 68)(36 81)(37 94)(38 107)(39 64)(40 77)(41 90)(42 103)(43 60)(44 73)(45 86)(46 99)(47 112)(48 69)(49 82)(50 95)(51 108)(52 65)(53 78)(54 91)(55 104)(56 61)
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)(57,62)(58,61)(59,60)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88), (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)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111), (1,74)(2,87)(3,100)(4,57)(5,70)(6,83)(7,96)(8,109)(9,66)(10,79)(11,92)(12,105)(13,62)(14,75)(15,88)(16,101)(17,58)(18,71)(19,84)(20,97)(21,110)(22,67)(23,80)(24,93)(25,106)(26,63)(27,76)(28,89)(29,102)(30,59)(31,72)(32,85)(33,98)(34,111)(35,68)(36,81)(37,94)(38,107)(39,64)(40,77)(41,90)(42,103)(43,60)(44,73)(45,86)(46,99)(47,112)(48,69)(49,82)(50,95)(51,108)(52,65)(53,78)(54,91)(55,104)(56,61)>;
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)(57,62)(58,61)(59,60)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88), (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)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111), (1,74)(2,87)(3,100)(4,57)(5,70)(6,83)(7,96)(8,109)(9,66)(10,79)(11,92)(12,105)(13,62)(14,75)(15,88)(16,101)(17,58)(18,71)(19,84)(20,97)(21,110)(22,67)(23,80)(24,93)(25,106)(26,63)(27,76)(28,89)(29,102)(30,59)(31,72)(32,85)(33,98)(34,111)(35,68)(36,81)(37,94)(38,107)(39,64)(40,77)(41,90)(42,103)(43,60)(44,73)(45,86)(46,99)(47,112)(48,69)(49,82)(50,95)(51,108)(52,65)(53,78)(54,91)(55,104)(56,61) );
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),(57,62),(58,61),(59,60),(63,112),(64,111),(65,110),(66,109),(67,108),(68,107),(69,106),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88)], [(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),(57,85),(59,87),(61,89),(63,91),(65,93),(67,95),(69,97),(71,99),(73,101),(75,103),(77,105),(79,107),(81,109),(83,111)], [(1,74),(2,87),(3,100),(4,57),(5,70),(6,83),(7,96),(8,109),(9,66),(10,79),(11,92),(12,105),(13,62),(14,75),(15,88),(16,101),(17,58),(18,71),(19,84),(20,97),(21,110),(22,67),(23,80),(24,93),(25,106),(26,63),(27,76),(28,89),(29,102),(30,59),(31,72),(32,85),(33,98),(34,111),(35,68),(36,81),(37,94),(38,107),(39,64),(40,77),(41,90),(42,103),(43,60),(44,73),(45,86),(46,99),(47,112),(48,69),(49,82),(50,95),(51,108),(52,65),(53,78),(54,91),(55,104),(56,61)])
Matrix representation ►G ⊆ 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] >;
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 |
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