metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊11C4, C8.14D28, C56.67D4, Dic28⋊11C4, C42.267D14, (C4×C8)⋊10D7, C7⋊1(C8○D8), (C4×C56)⋊15C2, C56⋊C2⋊7C4, C8.22(C4×D7), C14.9(C4×D4), C56.52(C2×C4), C4.76(C2×D28), C2.12(C4×D28), D28.13(C2×C4), (C2×C8).324D14, C28.296(C2×D4), C56.C4⋊17C2, Dic14⋊C4⋊15C2, D56⋊7C2.10C2, D28.2C4⋊11C2, (C4×C28).328C22, C28.104(C22×C4), (C2×C28).791C23, (C2×C56).407C22, Dic14.13(C2×C4), C4○D28.35C22, C22.20(C4○D28), C4.Dic7.33C22, C4.62(C2×C4×D7), (C2×C14).62(C4○D4), (C2×C4).672(C22×D7), SmallGroup(448,234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊11C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, ac=ca, cbc-1=a42b >
Subgroups: 484 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic7, C28, C28, D14, C2×C14, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C8○D8, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, C4×C28, C2×C56, C4○D28, Dic14⋊C4, C56.C4, C4×C56, D28.2C4, D56⋊7C2, D56⋊11C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, D28, C22×D7, C8○D8, C2×C4×D7, C2×D28, C4○D28, C4×D28, D56⋊11C4
►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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 71 85 99)(58 72 86 100)(59 73 87 101)(60 74 88 102)(61 75 89 103)(62 76 90 104)(63 77 91 105)(64 78 92 106)(65 79 93 107)(66 80 94 108)(67 81 95 109)(68 82 96 110)(69 83 97 111)(70 84 98 112)
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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,71,85,99)(58,72,86,100)(59,73,87,101)(60,74,88,102)(61,75,89,103)(62,76,90,104)(63,77,91,105)(64,78,92,106)(65,79,93,107)(66,80,94,108)(67,81,95,109)(68,82,96,110)(69,83,97,111)(70,84,98,112)>;
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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,71,85,99)(58,72,86,100)(59,73,87,101)(60,74,88,102)(61,75,89,103)(62,76,90,104)(63,77,91,105)(64,78,92,106)(65,79,93,107)(66,80,94,108)(67,81,95,109)(68,82,96,110)(69,83,97,111)(70,84,98,112) );
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)], [(1,100),(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,112),(46,111),(47,110),(48,109),(49,108),(50,107),(51,106),(52,105),(53,104),(54,103),(55,102),(56,101)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,71,85,99),(58,72,86,100),(59,73,87,101),(60,74,88,102),(61,75,89,103),(62,76,90,104),(63,77,91,105),(64,78,92,106),(65,79,93,107),(66,80,94,108),(67,81,95,109),(68,82,96,110),(69,83,97,111),(70,84,98,112)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C8○D8 | C4○D28 | D56⋊11C4 |
| kernel | D56⋊11C4 | Dic14⋊C4 | C56.C4 | C4×C56 | D28.2C4 | D56⋊7C2 | C56⋊C2 | D56 | Dic28 | C56 | C4×C8 | C2×C14 | C42 | C2×C8 | C8 | C8 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 3 | 2 | 3 | 6 | 12 | 12 | 8 | 12 | 48 |
Matrix representation of D56⋊11C4 ►in GL2(𝔽113) generated by
| 104 | 0 |
| 0 | 25 |
| 0 | 25 |
| 104 | 0 |
| 112 | 0 |
| 0 | 15 |
G:=sub<GL(2,GF(113))| [104,0,0,25],[0,104,25,0],[112,0,0,15] >;
D56⋊11C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes_{11}C_4 % in TeX
G:=Group("D56:11C4"); // GroupNames label
G:=SmallGroup(448,234);
// by ID
G=gap.SmallGroup(448,234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,58,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^42*b>;
// generators/relations