metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.56D14, C22≀C2⋊9D7, D14⋊C4⋊8C22, (C2×Dic7)⋊20D4, (D4×Dic7)⋊10C2, C24⋊D7⋊4C2, C22.39(D4×D7), Dic7⋊D4⋊1C2, Dic7⋊4D4⋊1C2, (C2×D4).148D14, (C2×C28).25C23, Dic7⋊C4⋊6C22, C4⋊Dic7⋊23C22, C22⋊C4.44D14, Dic7.44(C2×D4), (C23×Dic7)⋊5C2, C14.53(C22×D4), C22⋊3(D4⋊2D7), (C2×C14).130C24, C7⋊3(C22.19C24), (C4×Dic7)⋊12C22, C22.D28⋊8C2, (C22×C14).7C23, C23.D7⋊11C22, C22⋊Dic14⋊11C2, (C2×Dic14)⋊18C22, (D4×C14).109C22, C23.18D14⋊2C2, C23.11D14⋊1C2, (C23×C14).66C22, (C22×D7).52C23, C23.175(C22×D7), C22.151(C23×D7), (C2×Dic7).219C23, (C22×Dic7)⋊10C22, C2.26(C2×D4×D7), (C2×C4×D7)⋊4C22, (C7×C22≀C2)⋊2C2, (C2×D4⋊2D7)⋊5C2, (C2×C14)⋊9(C4○D4), C14.75(C2×C4○D4), (C2×C14).52(C2×D4), (C2×C7⋊D4)⋊6C22, C2.26(C2×D4⋊2D7), (C2×C4).25(C22×D7), (C7×C22⋊C4).1C22, SmallGroup(448,1039)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.56D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=d, ab=ba, ac=ca, eae-1=faf-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 1356 in 330 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, D4×C14, D4×C14, C23×C14, C23.11D14, C22⋊Dic14, Dic7⋊4D4, C22.D28, D4×Dic7, C23.18D14, Dic7⋊D4, C24⋊D7, C7×C22≀C2, C2×D4⋊2D7, C23×Dic7, C24.56D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C22×D7, C22.19C24, D4×D7, D4⋊2D7, C23×D7, C2×D4×D7, C2×D4⋊2D7, C24.56D14
►On 112 points
Generators in S112
(2 34)(4 36)(6 38)(8 40)(10 42)(12 30)(14 32)(16 88)(18 90)(20 92)(22 94)(24 96)(26 98)(28 86)(44 57)(46 59)(48 61)(50 63)(52 65)(54 67)(56 69)(71 112)(73 100)(75 102)(77 104)(79 106)(81 108)(83 110)
(2 61)(4 63)(6 65)(8 67)(10 69)(12 57)(14 59)(16 108)(18 110)(20 112)(22 100)(24 102)(26 104)(28 106)(30 44)(32 46)(34 48)(36 50)(38 52)(40 54)(42 56)(71 92)(73 94)(75 96)(77 98)(79 86)(81 88)(83 90)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 57)(13 58)(14 59)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 29)(12 30)(13 31)(14 32)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 85)(28 86)(43 70)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(71 112)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)
(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 92 33 20)(2 91 34 19)(3 90 35 18)(4 89 36 17)(5 88 37 16)(6 87 38 15)(7 86 39 28)(8 85 40 27)(9 98 41 26)(10 97 42 25)(11 96 29 24)(12 95 30 23)(13 94 31 22)(14 93 32 21)(43 102 70 75)(44 101 57 74)(45 100 58 73)(46 99 59 72)(47 112 60 71)(48 111 61 84)(49 110 62 83)(50 109 63 82)(51 108 64 81)(52 107 65 80)(53 106 66 79)(54 105 67 78)(55 104 68 77)(56 103 69 76)
G:=sub<Sym(112)| (2,34)(4,36)(6,38)(8,40)(10,42)(12,30)(14,32)(16,88)(18,90)(20,92)(22,94)(24,96)(26,98)(28,86)(44,57)(46,59)(48,61)(50,63)(52,65)(54,67)(56,69)(71,112)(73,100)(75,102)(77,104)(79,106)(81,108)(83,110), (2,61)(4,63)(6,65)(8,67)(10,69)(12,57)(14,59)(16,108)(18,110)(20,112)(22,100)(24,102)(26,104)(28,106)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56)(71,92)(73,94)(75,96)(77,98)(79,86)(81,88)(83,90), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,57)(13,58)(14,59)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,29)(12,30)(13,31)(14,32)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,85)(28,86)(43,70)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(71,112)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111), (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), (1,92,33,20)(2,91,34,19)(3,90,35,18)(4,89,36,17)(5,88,37,16)(6,87,38,15)(7,86,39,28)(8,85,40,27)(9,98,41,26)(10,97,42,25)(11,96,29,24)(12,95,30,23)(13,94,31,22)(14,93,32,21)(43,102,70,75)(44,101,57,74)(45,100,58,73)(46,99,59,72)(47,112,60,71)(48,111,61,84)(49,110,62,83)(50,109,63,82)(51,108,64,81)(52,107,65,80)(53,106,66,79)(54,105,67,78)(55,104,68,77)(56,103,69,76)>;
G:=Group( (2,34)(4,36)(6,38)(8,40)(10,42)(12,30)(14,32)(16,88)(18,90)(20,92)(22,94)(24,96)(26,98)(28,86)(44,57)(46,59)(48,61)(50,63)(52,65)(54,67)(56,69)(71,112)(73,100)(75,102)(77,104)(79,106)(81,108)(83,110), (2,61)(4,63)(6,65)(8,67)(10,69)(12,57)(14,59)(16,108)(18,110)(20,112)(22,100)(24,102)(26,104)(28,106)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56)(71,92)(73,94)(75,96)(77,98)(79,86)(81,88)(83,90), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,57)(13,58)(14,59)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,29)(12,30)(13,31)(14,32)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,85)(28,86)(43,70)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(71,112)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111), (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), (1,92,33,20)(2,91,34,19)(3,90,35,18)(4,89,36,17)(5,88,37,16)(6,87,38,15)(7,86,39,28)(8,85,40,27)(9,98,41,26)(10,97,42,25)(11,96,29,24)(12,95,30,23)(13,94,31,22)(14,93,32,21)(43,102,70,75)(44,101,57,74)(45,100,58,73)(46,99,59,72)(47,112,60,71)(48,111,61,84)(49,110,62,83)(50,109,63,82)(51,108,64,81)(52,107,65,80)(53,106,66,79)(54,105,67,78)(55,104,68,77)(56,103,69,76) );
G=PermutationGroup([[(2,34),(4,36),(6,38),(8,40),(10,42),(12,30),(14,32),(16,88),(18,90),(20,92),(22,94),(24,96),(26,98),(28,86),(44,57),(46,59),(48,61),(50,63),(52,65),(54,67),(56,69),(71,112),(73,100),(75,102),(77,104),(79,106),(81,108),(83,110)], [(2,61),(4,63),(6,65),(8,67),(10,69),(12,57),(14,59),(16,108),(18,110),(20,112),(22,100),(24,102),(26,104),(28,106),(30,44),(32,46),(34,48),(36,50),(38,52),(40,54),(42,56),(71,92),(73,94),(75,96),(77,98),(79,86),(81,88),(83,90)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,57),(13,58),(14,59),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,29),(12,30),(13,31),(14,32),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,85),(28,86),(43,70),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(71,112),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111)], [(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)], [(1,92,33,20),(2,91,34,19),(3,90,35,18),(4,89,36,17),(5,88,37,16),(6,87,38,15),(7,86,39,28),(8,85,40,27),(9,98,41,26),(10,97,42,25),(11,96,29,24),(12,95,30,23),(13,94,31,22),(14,93,32,21),(43,102,70,75),(44,101,57,74),(45,100,58,73),(46,99,59,72),(47,112,60,71),(48,111,61,84),(49,110,62,83),(50,109,63,82),(51,108,64,81),(52,107,65,80),(53,106,66,79),(54,105,67,78),(55,104,68,77),(56,103,69,76)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 28 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
70 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 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D4×D7 | D4⋊2D7 |
| kernel | C24.56D14 | C23.11D14 | C22⋊Dic14 | Dic7⋊4D4 | C22.D28 | D4×Dic7 | C23.18D14 | Dic7⋊D4 | C24⋊D7 | C7×C22≀C2 | C2×D4⋊2D7 | C23×Dic7 | C2×Dic7 | C22≀C2 | C2×C14 | C22⋊C4 | C2×D4 | C24 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 9 | 9 | 3 | 6 | 12 |
Matrix representation of C24.56D14 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 18 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 21 | 0 | 0 | 0 | 0 |
| 15 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,18,0,0,0,0,8,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,15,0,0,0,0,21,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;
C24.56D14 in GAP, Magma, Sage, TeX
C_2^4._{56}D_{14} % in TeX
G:=Group("C2^4.56D14"); // GroupNames label
G:=SmallGroup(448,1039);
// by ID
G=gap.SmallGroup(448,1039);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations