direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D56⋊C2, C56⋊4C23, SD16⋊8D14, D28⋊2C23, C28.6C24, D56⋊20C22, C7⋊C8⋊2C23, (C2×C8)⋊10D14, C4.43(D4×D7), C8⋊4(C22×D7), (C2×D56)⋊26C2, (C2×Q8)⋊21D14, (C2×SD16)⋊4D7, (C4×D7).15D4, C28.81(C2×D4), (D4×D7)⋊6C22, Q8⋊D7⋊8C22, (C7×Q8)⋊2C23, Q8⋊2(C22×D7), C4.6(C23×D7), C14⋊3(C8⋊C22), (C2×C56)⋊13C22, D4⋊D7⋊10C22, (C14×SD16)⋊5C2, D14.50(C2×D4), C8⋊D7⋊8C22, (C4×D7).3C23, (C7×D4).4C23, D4.4(C22×D7), (C2×D4).182D14, (C2×D28)⋊33C22, Dic7.55(C2×D4), (Q8×C14)⋊18C22, Q8⋊2D7⋊5C22, (C7×SD16)⋊8C22, (C22×D7).98D4, C22.139(D4×D7), (C2×C28).523C23, (C2×Dic7).192D4, C14.107(C22×D4), (D4×C14).164C22, (C2×D4×D7)⋊23C2, C7⋊3(C2×C8⋊C22), C2.80(C2×D4×D7), (C2×D4⋊D7)⋊27C2, (C2×C8⋊D7)⋊4C2, (C2×C7⋊C8)⋊15C22, (C2×Q8⋊D7)⋊26C2, (C2×Q8⋊2D7)⋊14C2, (C2×C14).396(C2×D4), (C2×C4×D7).156C22, (C2×C4).612(C22×D7), SmallGroup(448,1212)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1796 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×2], D4 [×15], Q8 [×2], Q8, C23 [×12], D7 [×6], C14, C14 [×2], C14 [×2], C2×C8, C2×C8, M4(2) [×4], D8 [×8], SD16 [×4], SD16 [×4], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×6], C24, Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×18], C2×C14, C2×C14 [×4], C2×M4(2), C2×D8 [×2], C2×SD16, C2×SD16, C8⋊C22 [×8], C22×D4, C2×C4○D4, C7⋊C8 [×2], C56 [×2], C4×D7 [×4], C4×D7 [×4], D28 [×4], D28 [×6], C2×Dic7, C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C7×Q8 [×2], C7×Q8, C22×D7, C22×D7 [×10], C22×C14, C2×C8⋊C22, C8⋊D7 [×4], D56 [×4], C2×C7⋊C8, D4⋊D7 [×4], Q8⋊D7 [×4], C2×C56, C7×SD16 [×4], C2×C4×D7, C2×C4×D7, C2×D28 [×2], C2×D28, D4×D7 [×4], D4×D7 [×2], Q8⋊2D7 [×4], Q8⋊2D7 [×2], C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C2×C8⋊D7, C2×D56, D56⋊C2 [×8], C2×D4⋊D7, C2×Q8⋊D7, C14×SD16, C2×D4×D7, C2×Q8⋊2D7, C2×D56⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8⋊C22 [×2], C22×D4, C22×D7 [×7], C2×C8⋊C22, D4×D7 [×2], C23×D7, D56⋊C2 [×2], C2×D4×D7, C2×D56⋊C2
Generators and relations
G = < a,b,c,d | a2=b56=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b43, cd=dc >
►On 112 points
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 82)(2 69)(3 112)(4 99)(5 86)(6 73)(7 60)(8 103)(9 90)(10 77)(11 64)(12 107)(13 94)(14 81)(15 68)(16 111)(17 98)(18 85)(19 72)(20 59)(21 102)(22 89)(23 76)(24 63)(25 106)(26 93)(27 80)(28 67)(29 110)(30 97)(31 84)(32 71)(33 58)(34 101)(35 88)(36 75)(37 62)(38 105)(39 92)(40 79)(41 66)(42 109)(43 96)(44 83)(45 70)(46 57)(47 100)(48 87)(49 74)(50 61)(51 104)(52 91)(53 78)(54 65)(55 108)(56 95)
G:=sub<Sym(112)| (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,82)(2,69)(3,112)(4,99)(5,86)(6,73)(7,60)(8,103)(9,90)(10,77)(11,64)(12,107)(13,94)(14,81)(15,68)(16,111)(17,98)(18,85)(19,72)(20,59)(21,102)(22,89)(23,76)(24,63)(25,106)(26,93)(27,80)(28,67)(29,110)(30,97)(31,84)(32,71)(33,58)(34,101)(35,88)(36,75)(37,62)(38,105)(39,92)(40,79)(41,66)(42,109)(43,96)(44,83)(45,70)(46,57)(47,100)(48,87)(49,74)(50,61)(51,104)(52,91)(53,78)(54,65)(55,108)(56,95)>;
G:=Group( (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,82)(2,69)(3,112)(4,99)(5,86)(6,73)(7,60)(8,103)(9,90)(10,77)(11,64)(12,107)(13,94)(14,81)(15,68)(16,111)(17,98)(18,85)(19,72)(20,59)(21,102)(22,89)(23,76)(24,63)(25,106)(26,93)(27,80)(28,67)(29,110)(30,97)(31,84)(32,71)(33,58)(34,101)(35,88)(36,75)(37,62)(38,105)(39,92)(40,79)(41,66)(42,109)(43,96)(44,83)(45,70)(46,57)(47,100)(48,87)(49,74)(50,61)(51,104)(52,91)(53,78)(54,65)(55,108)(56,95) );
G=PermutationGroup([(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,82),(2,69),(3,112),(4,99),(5,86),(6,73),(7,60),(8,103),(9,90),(10,77),(11,64),(12,107),(13,94),(14,81),(15,68),(16,111),(17,98),(18,85),(19,72),(20,59),(21,102),(22,89),(23,76),(24,63),(25,106),(26,93),(27,80),(28,67),(29,110),(30,97),(31,84),(32,71),(33,58),(34,101),(35,88),(36,75),(37,62),(38,105),(39,92),(40,79),(41,66),(42,109),(43,96),(44,83),(45,70),(46,57),(47,100),(48,87),(49,74),(50,61),(51,104),(52,91),(53,78),(54,65),(55,108),(56,95)])
Matrix representation ►G ⊆ GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 100 | 1 | 22 | 92 | 0 | 0 | 0 | 0 |
| 23 | 38 | 86 | 98 | 0 | 0 | 0 | 0 |
| 93 | 84 | 66 | 112 | 0 | 0 | 0 | 0 |
| 42 | 15 | 65 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 44 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 76 | 28 |
| 0 | 0 | 0 | 0 | 91 | 38 | 69 | 76 |
| 0 | 0 | 0 | 0 | 75 | 99 | 37 | 85 |
| 9 | 89 | 0 | 0 | 0 | 0 | 0 | 0 |
| 41 | 104 | 0 | 0 | 0 | 0 | 0 | 0 |
| 69 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 57 | 74 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 111 |
| 0 | 0 | 0 | 0 | 112 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 83 | 112 | 0 | 0 | 0 | 0 | 0 |
| 101 | 44 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[100,23,93,42,0,0,0,0,1,38,84,15,0,0,0,0,22,86,66,65,0,0,0,0,92,98,112,22,0,0,0,0,0,0,0,0,0,0,91,75,0,0,0,0,0,0,38,99,0,0,0,0,44,76,69,37,0,0,0,0,37,28,76,85],[9,41,69,57,0,0,0,0,89,104,0,74,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,111,0,1,0,0,0,0,111,0,1,0],[1,0,0,101,0,0,0,0,0,1,83,44,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,112,0,0,0,0,0,0,2,0,112] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C8⋊C22 | D4×D7 | D4×D7 | D56⋊C2 |
| kernel | C2×D56⋊C2 | C2×C8⋊D7 | C2×D56 | D56⋊C2 | C2×D4⋊D7 | C2×Q8⋊D7 | C14×SD16 | C2×D4×D7 | C2×Q8⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 12 | 3 | 3 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_{56}\rtimes C_2 % in TeX
G:=Group("C2xD56:C2"); // GroupNames label
G:=SmallGroup(448,1212);
// by ID
G=gap.SmallGroup(448,1212);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^56=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^43,c*d=d*c>;
// generators/relations