metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.49C23, C28.72C24, M4(2)⋊28D14, (D4×D7).C4, (Q8×D7).C4, C8○D4⋊8D7, (C2×C8)⋊23D14, D4⋊2D7.C4, Q8⋊2D7.C4, D4.13(C4×D7), C7⋊C8.33C23, Q8.14(C4×D7), D28.C4⋊13C2, (C2×C56)⋊26C22, C4○D4.43D14, D28.21(C2×C4), (C8×D7)⋊12C22, C7⋊2(Q8○M4(2)), D4.Dic7⋊8C2, C8.56(C22×D7), C4.71(C23×D7), C8⋊D7⋊22C22, (D7×M4(2))⋊11C2, C14.35(C23×C4), C28.39(C22×C4), (C4×D7).37C23, D28.2C4⋊17C2, (C2×C28).514C23, Dic14.22(C2×C4), C4○D28.52C22, D14.16(C22×C4), C4.Dic7⋊27C22, (C7×M4(2))⋊28C22, Dic7.16(C22×C4), C4.39(C2×C4×D7), (C7×C8○D4)⋊9C2, C22.5(C2×C4×D7), (C2×C7⋊C8)⋊13C22, (D7×C4○D4).3C2, C7⋊D4.2(C2×C4), (C2×C8⋊D7)⋊28C2, C2.36(D7×C22×C4), (C4×D7).11(C2×C4), (C7×D4).17(C2×C4), (C7×Q8).18(C2×C4), (C2×C14).5(C22×C4), (C2×C4×D7).154C22, (C2×Dic7).38(C2×C4), (C7×C4○D4).44C22, (C22×D7).28(C2×C4), (C2×C4).607(C22×D7), SmallGroup(448,1203)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 956 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C7, C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D7 [×4], C14, C14 [×3], C2×C8 [×3], C2×C8 [×9], M4(2) [×3], M4(2) [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic7, Dic7 [×3], C28, C28 [×3], D14, D14 [×3], D14 [×3], C2×C14 [×3], C2×M4(2) [×6], C8○D4, C8○D4 [×7], C2×C4○D4, C7⋊C8, C7⋊C8 [×3], C56, C56 [×3], Dic14 [×3], C4×D7, C4×D7 [×9], D28 [×3], C2×Dic7 [×3], C7⋊D4 [×6], C2×C28 [×3], C7×D4 [×3], C7×Q8, C22×D7 [×3], Q8○M4(2), C8×D7 [×6], C8⋊D7, C8⋊D7 [×9], C2×C7⋊C8 [×3], C4.Dic7 [×3], C2×C56 [×3], C7×M4(2) [×3], C2×C4×D7 [×3], C4○D28 [×3], D4×D7 [×3], D4⋊2D7 [×3], Q8×D7, Q8⋊2D7, C7×C4○D4, C2×C8⋊D7 [×3], D28.2C4 [×3], D7×M4(2) [×3], D28.C4 [×3], D4.Dic7, C7×C8○D4, D7×C4○D4, C56.49C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, D14 [×7], C23×C4, C4×D7 [×4], C22×D7 [×7], Q8○M4(2), C2×C4×D7 [×6], C23×D7, D7×C22×C4, C56.49C23
Generators and relations
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a13, cac=a29, ad=da, bc=cb, bd=db, dcd=a28c >
►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)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 93)(58 106)(59 63)(60 76)(61 89)(62 102)(64 72)(65 85)(66 98)(67 111)(69 81)(70 94)(71 107)(73 77)(74 90)(75 103)(78 86)(79 99)(80 112)(83 95)(84 108)(87 91)(88 104)(92 100)(97 109)(101 105)
(1 68)(2 97)(3 70)(4 99)(5 72)(6 101)(7 74)(8 103)(9 76)(10 105)(11 78)(12 107)(13 80)(14 109)(15 82)(16 111)(17 84)(18 57)(19 86)(20 59)(21 88)(22 61)(23 90)(24 63)(25 92)(26 65)(27 94)(28 67)(29 96)(30 69)(31 98)(32 71)(33 100)(34 73)(35 102)(36 75)(37 104)(38 77)(39 106)(40 79)(41 108)(42 81)(43 110)(44 83)(45 112)(46 85)(47 58)(48 87)(49 60)(50 89)(51 62)(52 91)(53 64)(54 93)(55 66)(56 95)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
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), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,93)(58,106)(59,63)(60,76)(61,89)(62,102)(64,72)(65,85)(66,98)(67,111)(69,81)(70,94)(71,107)(73,77)(74,90)(75,103)(78,86)(79,99)(80,112)(83,95)(84,108)(87,91)(88,104)(92,100)(97,109)(101,105), (1,68)(2,97)(3,70)(4,99)(5,72)(6,101)(7,74)(8,103)(9,76)(10,105)(11,78)(12,107)(13,80)(14,109)(15,82)(16,111)(17,84)(18,57)(19,86)(20,59)(21,88)(22,61)(23,90)(24,63)(25,92)(26,65)(27,94)(28,67)(29,96)(30,69)(31,98)(32,71)(33,100)(34,73)(35,102)(36,75)(37,104)(38,77)(39,106)(40,79)(41,108)(42,81)(43,110)(44,83)(45,112)(46,85)(47,58)(48,87)(49,60)(50,89)(51,62)(52,91)(53,64)(54,93)(55,66)(56,95), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)>;
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), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,93)(58,106)(59,63)(60,76)(61,89)(62,102)(64,72)(65,85)(66,98)(67,111)(69,81)(70,94)(71,107)(73,77)(74,90)(75,103)(78,86)(79,99)(80,112)(83,95)(84,108)(87,91)(88,104)(92,100)(97,109)(101,105), (1,68)(2,97)(3,70)(4,99)(5,72)(6,101)(7,74)(8,103)(9,76)(10,105)(11,78)(12,107)(13,80)(14,109)(15,82)(16,111)(17,84)(18,57)(19,86)(20,59)(21,88)(22,61)(23,90)(24,63)(25,92)(26,65)(27,94)(28,67)(29,96)(30,69)(31,98)(32,71)(33,100)(34,73)(35,102)(36,75)(37,104)(38,77)(39,106)(40,79)(41,108)(42,81)(43,110)(44,83)(45,112)(46,85)(47,58)(48,87)(49,60)(50,89)(51,62)(52,91)(53,64)(54,93)(55,66)(56,95), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112) );
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)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,93),(58,106),(59,63),(60,76),(61,89),(62,102),(64,72),(65,85),(66,98),(67,111),(69,81),(70,94),(71,107),(73,77),(74,90),(75,103),(78,86),(79,99),(80,112),(83,95),(84,108),(87,91),(88,104),(92,100),(97,109),(101,105)], [(1,68),(2,97),(3,70),(4,99),(5,72),(6,101),(7,74),(8,103),(9,76),(10,105),(11,78),(12,107),(13,80),(14,109),(15,82),(16,111),(17,84),(18,57),(19,86),(20,59),(21,88),(22,61),(23,90),(24,63),(25,92),(26,65),(27,94),(28,67),(29,96),(30,69),(31,98),(32,71),(33,100),(34,73),(35,102),(36,75),(37,104),(38,77),(39,106),(40,79),(41,108),(42,81),(43,110),(44,83),(45,112),(46,85),(47,58),(48,87),(49,60),(50,89),(51,62),(52,91),(53,64),(54,93),(55,66),(56,95)], [(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 76 | 49 | 0 | 0 |
| 64 | 105 | 0 | 0 |
| 0 | 0 | 37 | 64 |
| 0 | 0 | 49 | 8 |
| 1 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 79 | 112 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [76,64,0,0,49,105,0,0,0,0,37,49,0,0,64,8],[1,79,0,0,0,112,0,0,0,0,1,79,0,0,0,112],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112] >;
94 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 | ··· | 8H | 8I | ··· | 8P | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 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 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | D14 | D14 | D14 | C4×D7 | C4×D7 | Q8○M4(2) | C56.49C23 |
| kernel | C56.49C23 | C2×C8⋊D7 | D28.2C4 | D7×M4(2) | D28.C4 | D4.Dic7 | C7×C8○D4 | D7×C4○D4 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 3 | 9 | 9 | 3 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{56}._{49}C_2^3 % in TeX
G:=Group("C56.49C2^3"); // GroupNames label
G:=SmallGroup(448,1203);
// by ID
G=gap.SmallGroup(448,1203);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^56=b^2=c^2=d^2=1,b*a*b=a^13,c*a*c=a^29,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^28*c>;
// generators/relations