metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.47C23, C28.70C24, M4(2)⋊26D14, (C2×C8)⋊22D14, C4○D28.5C4, C7⋊C8.32C23, D28.C4⋊12C2, (C2×C56)⋊24C22, D28.29(C2×C4), (C2×D28).17C4, (C8×D7)⋊11C22, C7⋊1(Q8○M4(2)), C23.22(C4×D7), C8.44(C22×D7), C4.69(C23×D7), C8⋊D7⋊19C22, (D7×M4(2))⋊10C2, (C2×M4(2))⋊16D7, C14.33(C23×C4), C28.93(C22×C4), (C4×D7).36C23, D28.2C4⋊15C2, (C14×M4(2))⋊10C2, (C2×C28).510C23, Dic14.30(C2×C4), (C2×Dic14).17C4, C4○D28.59C22, D14.14(C22×C4), (C22×C4).264D14, C4.Dic7⋊40C22, (C7×M4(2))⋊26C22, Dic7.14(C22×C4), (C22×C28).265C22, C4.95(C2×C4×D7), (C2×C7⋊C8)⋊12C22, (C2×C4).58(C4×D7), C7⋊D4.4(C2×C4), C2.34(D7×C22×C4), C22.28(C2×C4×D7), (C4×D7).10(C2×C4), (C2×C7⋊D4).15C4, (C2×C28).132(C2×C4), (C2×C4○D28).22C2, (C2×C4.Dic7)⋊25C2, (C2×C4×D7).153C22, (C22×C14).79(C2×C4), (C2×C14).26(C22×C4), (C2×Dic7).37(C2×C4), (C22×D7).27(C2×C4), (C2×C4).605(C22×D7), SmallGroup(448,1198)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 932 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×7], C7, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D7 [×4], C14, C14 [×3], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic7 [×4], C28 [×4], D14 [×4], D14 [×2], C2×C14 [×3], C2×C14, C2×M4(2), C2×M4(2) [×5], C8○D4 [×8], C2×C4○D4, C7⋊C8 [×4], C56 [×4], Dic14 [×4], C4×D7 [×8], D28 [×4], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×6], C22×D7 [×2], C22×C14, Q8○M4(2), C8×D7 [×8], C8⋊D7 [×8], C2×C7⋊C8 [×2], C4.Dic7 [×4], C2×C56 [×2], C7×M4(2) [×4], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×8], C2×C7⋊D4 [×2], C22×C28, D28.2C4 [×4], D7×M4(2) [×4], D28.C4 [×4], C2×C4.Dic7, C14×M4(2), C2×C4○D28, C56.47C23
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, C56.47C23
Generators and relations
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a41, ac=ca, dad=a29, cbc=a28b, bd=db, 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)
(2 42)(3 27)(4 12)(5 53)(6 38)(7 23)(9 49)(10 34)(11 19)(13 45)(14 30)(16 56)(17 41)(18 26)(20 52)(21 37)(24 48)(25 33)(28 44)(31 55)(32 40)(35 51)(39 47)(46 54)(57 93)(58 78)(59 63)(60 104)(61 89)(62 74)(64 100)(65 85)(66 70)(67 111)(68 96)(69 81)(71 107)(72 92)(73 77)(75 103)(76 88)(79 99)(80 84)(82 110)(83 95)(86 106)(87 91)(90 102)(94 98)(97 109)(101 105)(108 112)
(1 110)(2 111)(3 112)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)
(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)
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), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,93)(58,78)(59,63)(60,104)(61,89)(62,74)(64,100)(65,85)(66,70)(67,111)(68,96)(69,81)(71,107)(72,92)(73,77)(75,103)(76,88)(79,99)(80,84)(82,110)(83,95)(86,106)(87,91)(90,102)(94,98)(97,109)(101,105)(108,112), (1,110)(2,111)(3,112)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109), (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)>;
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), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,93)(58,78)(59,63)(60,104)(61,89)(62,74)(64,100)(65,85)(66,70)(67,111)(68,96)(69,81)(71,107)(72,92)(73,77)(75,103)(76,88)(79,99)(80,84)(82,110)(83,95)(86,106)(87,91)(90,102)(94,98)(97,109)(101,105)(108,112), (1,110)(2,111)(3,112)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109), (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) );
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)], [(2,42),(3,27),(4,12),(5,53),(6,38),(7,23),(9,49),(10,34),(11,19),(13,45),(14,30),(16,56),(17,41),(18,26),(20,52),(21,37),(24,48),(25,33),(28,44),(31,55),(32,40),(35,51),(39,47),(46,54),(57,93),(58,78),(59,63),(60,104),(61,89),(62,74),(64,100),(65,85),(66,70),(67,111),(68,96),(69,81),(71,107),(72,92),(73,77),(75,103),(76,88),(79,99),(80,84),(82,110),(83,95),(86,106),(87,91),(90,102),(94,98),(97,109),(101,105),(108,112)], [(1,110),(2,111),(3,112),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109)], [(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)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 104 | 1 | 15 | 92 |
| 41 | 33 | 43 | 0 |
| 97 | 0 | 0 | 112 |
| 31 | 97 | 1 | 89 |
| 0 | 89 | 0 | 0 |
| 80 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 24 | 112 |
| 79 | 7 | 0 | 0 |
| 61 | 34 | 0 | 0 |
| 0 | 0 | 29 | 7 |
| 0 | 0 | 106 | 84 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 27 | 30 | 112 | 0 |
| 83 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [104,41,97,31,1,33,0,97,15,43,0,1,92,0,112,89],[0,80,0,0,89,0,0,0,0,0,1,24,0,0,0,112],[79,61,0,0,7,34,0,0,0,0,29,106,0,0,7,84],[1,0,27,83,0,1,30,0,0,0,112,0,0,0,0,112] >;
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 | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 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 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
94 irreducible representations
| dim | 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 | C4 | C4 | C4 | C4 | D7 | D14 | D14 | D14 | C4×D7 | C4×D7 | Q8○M4(2) | C56.47C23 |
| kernel | C56.47C23 | D28.2C4 | D7×M4(2) | D28.C4 | C2×C4.Dic7 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C7⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 3 | 6 | 12 | 3 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{56}._{47}C_2^3 % in TeX
G:=Group("C56.47C2^3"); // GroupNames label
G:=SmallGroup(448,1198);
// by ID
G=gap.SmallGroup(448,1198);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,570,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^56=b^2=c^2=d^2=1,b*a*b=a^41,a*c=c*a,d*a*d=a^29,c*b*c=a^28*b,b*d=d*b,c*d=d*c>;
// generators/relations