metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.42D14, C14.912+ 1+4, (C7×D4)⋊17D4, D4⋊8(C7⋊D4), C28⋊2D4⋊41C2, C7⋊10(D4⋊5D4), (D4×Dic7)⋊39C2, C28.253(C2×D4), (C22×D4)⋊12D7, D14⋊C4⋊73C22, (C2×D4).231D14, C24⋊D7⋊14C2, C4⋊Dic7⋊45C22, Dic7⋊D4⋊42C2, C28.48D4⋊37C2, C28.17D4⋊29C2, C22⋊5(D4⋊2D7), (C2×C28).546C23, (C2×C14).301C24, Dic7⋊C4⋊39C22, (C4×Dic7)⋊43C22, (C22×C4).273D14, C14.148(C22×D4), C23.D7⋊40C22, C2.94(D4⋊6D14), (C2×Dic14)⋊42C22, (D4×C14).312C22, (C23×C14).80C22, C22.314(C23×D7), C23.236(C22×D7), C23.18D14⋊30C2, (C22×C28).278C22, (C22×C14).235C23, (C2×Dic7).286C23, (C22×Dic7)⋊35C22, (C22×D7).132C23, (D4×C2×C14)⋊8C2, (C4×C7⋊D4)⋊26C2, (C2×C4×D7)⋊32C22, C4.68(C2×C7⋊D4), (C2×C14).74(C2×D4), (C2×D4⋊2D7)⋊27C2, (C2×C14)⋊15(C4○D4), C22.3(C2×C7⋊D4), C14.107(C2×C4○D4), C2.71(C2×D4⋊2D7), (C2×C7⋊D4)⋊29C22, (C2×C23.D7)⋊31C2, C2.21(C22×C7⋊D4), (C2×C4).239(C22×D7), SmallGroup(448,1259)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.42D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1332 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C22×C14, C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, D4×C14, C23×C14, C28.48D4, C4×C7⋊D4, D4×Dic7, C23.18D14, C28.17D4, C28⋊2D4, Dic7⋊D4, C2×C23.D7, C24⋊D7, C2×D4⋊2D7, D4×C2×C14, C24.42D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C7⋊D4, C22×D7, D4⋊5D4, D4⋊2D7, C2×C7⋊D4, C23×D7, C2×D4⋊2D7, D4⋊6D14, C22×C7⋊D4, C24.42D14
►On 112 points
Generators in S112
(1 43)(2 30)(3 45)(4 32)(5 47)(6 34)(7 49)(8 36)(9 51)(10 38)(11 53)(12 40)(13 55)(14 42)(15 29)(16 44)(17 31)(18 46)(19 33)(20 48)(21 35)(22 50)(23 37)(24 52)(25 39)(26 54)(27 41)(28 56)(57 103)(58 90)(59 105)(60 92)(61 107)(62 94)(63 109)(64 96)(65 111)(66 98)(67 85)(68 100)(69 87)(70 102)(71 89)(72 104)(73 91)(74 106)(75 93)(76 108)(77 95)(78 110)(79 97)(80 112)(81 99)(82 86)(83 101)(84 88)
(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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)
(1 101 15 87)(2 86 16 100)(3 99 17 85)(4 112 18 98)(5 97 19 111)(6 110 20 96)(7 95 21 109)(8 108 22 94)(9 93 23 107)(10 106 24 92)(11 91 25 105)(12 104 26 90)(13 89 27 103)(14 102 28 88)(29 69 43 83)(30 82 44 68)(31 67 45 81)(32 80 46 66)(33 65 47 79)(34 78 48 64)(35 63 49 77)(36 76 50 62)(37 61 51 75)(38 74 52 60)(39 59 53 73)(40 72 54 58)(41 57 55 71)(42 70 56 84)
G:=sub<Sym(112)| (1,43)(2,30)(3,45)(4,32)(5,47)(6,34)(7,49)(8,36)(9,51)(10,38)(11,53)(12,40)(13,55)(14,42)(15,29)(16,44)(17,31)(18,46)(19,33)(20,48)(21,35)(22,50)(23,37)(24,52)(25,39)(26,54)(27,41)(28,56)(57,103)(58,90)(59,105)(60,92)(61,107)(62,94)(63,109)(64,96)(65,111)(66,98)(67,85)(68,100)(69,87)(70,102)(71,89)(72,104)(73,91)(74,106)(75,93)(76,108)(77,95)(78,110)(79,97)(80,112)(81,99)(82,86)(83,101)(84,88), (57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,101,15,87)(2,86,16,100)(3,99,17,85)(4,112,18,98)(5,97,19,111)(6,110,20,96)(7,95,21,109)(8,108,22,94)(9,93,23,107)(10,106,24,92)(11,91,25,105)(12,104,26,90)(13,89,27,103)(14,102,28,88)(29,69,43,83)(30,82,44,68)(31,67,45,81)(32,80,46,66)(33,65,47,79)(34,78,48,64)(35,63,49,77)(36,76,50,62)(37,61,51,75)(38,74,52,60)(39,59,53,73)(40,72,54,58)(41,57,55,71)(42,70,56,84)>;
G:=Group( (1,43)(2,30)(3,45)(4,32)(5,47)(6,34)(7,49)(8,36)(9,51)(10,38)(11,53)(12,40)(13,55)(14,42)(15,29)(16,44)(17,31)(18,46)(19,33)(20,48)(21,35)(22,50)(23,37)(24,52)(25,39)(26,54)(27,41)(28,56)(57,103)(58,90)(59,105)(60,92)(61,107)(62,94)(63,109)(64,96)(65,111)(66,98)(67,85)(68,100)(69,87)(70,102)(71,89)(72,104)(73,91)(74,106)(75,93)(76,108)(77,95)(78,110)(79,97)(80,112)(81,99)(82,86)(83,101)(84,88), (57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,101,15,87)(2,86,16,100)(3,99,17,85)(4,112,18,98)(5,97,19,111)(6,110,20,96)(7,95,21,109)(8,108,22,94)(9,93,23,107)(10,106,24,92)(11,91,25,105)(12,104,26,90)(13,89,27,103)(14,102,28,88)(29,69,43,83)(30,82,44,68)(31,67,45,81)(32,80,46,66)(33,65,47,79)(34,78,48,64)(35,63,49,77)(36,76,50,62)(37,61,51,75)(38,74,52,60)(39,59,53,73)(40,72,54,58)(41,57,55,71)(42,70,56,84) );
G=PermutationGroup([[(1,43),(2,30),(3,45),(4,32),(5,47),(6,34),(7,49),(8,36),(9,51),(10,38),(11,53),(12,40),(13,55),(14,42),(15,29),(16,44),(17,31),(18,46),(19,33),(20,48),(21,35),(22,50),(23,37),(24,52),(25,39),(26,54),(27,41),(28,56),(57,103),(58,90),(59,105),(60,92),(61,107),(62,94),(63,109),(64,96),(65,111),(66,98),(67,85),(68,100),(69,87),(70,102),(71,89),(72,104),(73,91),(74,106),(75,93),(76,108),(77,95),(78,110),(79,97),(80,112),(81,99),(82,86),(83,101),(84,88)], [(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(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),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,101,15,87),(2,86,16,100),(3,99,17,85),(4,112,18,98),(5,97,19,111),(6,110,20,96),(7,95,21,109),(8,108,22,94),(9,93,23,107),(10,106,24,92),(11,91,25,105),(12,104,26,90),(13,89,27,103),(14,102,28,88),(29,69,43,83),(30,82,44,68),(31,67,45,81),(32,80,46,66),(33,65,47,79),(34,78,48,64),(35,63,49,77),(36,76,50,62),(37,61,51,75),(38,74,52,60),(39,59,53,73),(40,72,54,58),(41,57,55,71),(42,70,56,84)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 28 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2+ 1+4 | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.42D14 | C28.48D4 | C4×C7⋊D4 | D4×Dic7 | C23.18D14 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C2×C23.D7 | C24⋊D7 | C2×D4⋊2D7 | D4×C2×C14 | C7×D4 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | D4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 4 | 3 | 12 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C24.42D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 22 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[22,0,0,0,0,4,0,0,0,0,0,1,0,0,28,0],[0,22,0,0,4,0,0,0,0,0,12,0,0,0,0,12] >;
C24.42D14 in GAP, Magma, Sage, TeX
C_2^4._{42}D_{14} % in TeX
G:=Group("C2^4.42D14"); // GroupNames label
G:=SmallGroup(448,1259);
// by ID
G=gap.SmallGroup(448,1259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*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^13>;
// generators/relations