metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.30D14, C14.52+ 1+4, C28⋊7D4⋊4C2, D14⋊D4⋊2C2, D14⋊C4⋊2C22, (C2×D28)⋊4C22, C24⋊D7⋊3C2, C4⋊Dic7⋊6C22, D14.D4⋊2C2, C28.48D4⋊4C2, (C2×C14).38C24, Dic7⋊C4⋊2C22, C22⋊C4.87D14, (C22×C4).45D14, C2.9(D4⋊6D14), (C2×C28).131C23, Dic7.D4⋊2C2, C7⋊1(C22.32C24), (C2×Dic14)⋊3C22, (C4×Dic7)⋊48C22, C23.D14⋊1C2, C23.82(C22×D7), C22.77(C23×D7), C23.D7.2C22, C22.23(C4○D28), (C23×C14).64C22, (C2×Dic7).11C23, (C22×D7).10C23, (C22×C14).128C23, (C22×C28).355C22, (C4×C7⋊D4)⋊34C2, (C2×C4×D7)⋊41C22, (C2×C22⋊C4)⋊17D7, C2.18(C2×C4○D28), C14.16(C2×C4○D4), (C14×C22⋊C4)⋊20C2, (C2×C7⋊D4).7C22, (C2×C4).261(C22×D7), (C2×C14).104(C4○D4), (C7×C22⋊C4).109C22, SmallGroup(448,947)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.30D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1172 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×C14, C22×C14, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C22×C28, C23×C14, C23.D14, D14.D4, D14⋊D4, Dic7.D4, C28.48D4, C4×C7⋊D4, C28⋊7D4, C24⋊D7, C14×C22⋊C4, C24.30D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C4○D28, C23×D7, C2×C4○D28, D4⋊6D14, C24.30D14
►On 112 points
Generators in S112
(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 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 85)(82 86)(83 87)(84 88)
(2 50)(4 52)(6 54)(8 56)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(22 42)(24 44)(26 46)(28 48)(57 89)(58 72)(59 91)(60 74)(61 93)(62 76)(63 95)(64 78)(65 97)(66 80)(67 99)(68 82)(69 101)(70 84)(71 103)(73 105)(75 107)(77 109)(79 111)(81 85)(83 87)(86 100)(88 102)(90 104)(92 106)(94 108)(96 110)(98 112)
(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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(57 103)(58 104)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)
(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 87 15 101)(2 100 16 86)(3 85 17 99)(4 98 18 112)(5 111 19 97)(6 96 20 110)(7 109 21 95)(8 94 22 108)(9 107 23 93)(10 92 24 106)(11 105 25 91)(12 90 26 104)(13 103 27 89)(14 88 28 102)(29 61 43 75)(30 74 44 60)(31 59 45 73)(32 72 46 58)(33 57 47 71)(34 70 48 84)(35 83 49 69)(36 68 50 82)(37 81 51 67)(38 66 52 80)(39 79 53 65)(40 64 54 78)(41 77 55 63)(42 62 56 76)
G:=sub<Sym(112)| (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)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,85)(82,86)(83,87)(84,88), (2,50)(4,52)(6,54)(8,56)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(22,42)(24,44)(26,46)(28,48)(57,89)(58,72)(59,91)(60,74)(61,93)(62,76)(63,95)(64,78)(65,97)(66,80)(67,99)(68,82)(69,101)(70,84)(71,103)(73,105)(75,107)(77,109)(79,111)(81,85)(83,87)(86,100)(88,102)(90,104)(92,106)(94,108)(96,110)(98,112), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(57,103)(58,104)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102), (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), (1,87,15,101)(2,100,16,86)(3,85,17,99)(4,98,18,112)(5,111,19,97)(6,96,20,110)(7,109,21,95)(8,94,22,108)(9,107,23,93)(10,92,24,106)(11,105,25,91)(12,90,26,104)(13,103,27,89)(14,88,28,102)(29,61,43,75)(30,74,44,60)(31,59,45,73)(32,72,46,58)(33,57,47,71)(34,70,48,84)(35,83,49,69)(36,68,50,82)(37,81,51,67)(38,66,52,80)(39,79,53,65)(40,64,54,78)(41,77,55,63)(42,62,56,76)>;
G:=Group( (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)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,85)(82,86)(83,87)(84,88), (2,50)(4,52)(6,54)(8,56)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(22,42)(24,44)(26,46)(28,48)(57,89)(58,72)(59,91)(60,74)(61,93)(62,76)(63,95)(64,78)(65,97)(66,80)(67,99)(68,82)(69,101)(70,84)(71,103)(73,105)(75,107)(77,109)(79,111)(81,85)(83,87)(86,100)(88,102)(90,104)(92,106)(94,108)(96,110)(98,112), (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), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(57,103)(58,104)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102), (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), (1,87,15,101)(2,100,16,86)(3,85,17,99)(4,98,18,112)(5,111,19,97)(6,96,20,110)(7,109,21,95)(8,94,22,108)(9,107,23,93)(10,92,24,106)(11,105,25,91)(12,90,26,104)(13,103,27,89)(14,88,28,102)(29,61,43,75)(30,74,44,60)(31,59,45,73)(32,72,46,58)(33,57,47,71)(34,70,48,84)(35,83,49,69)(36,68,50,82)(37,81,51,67)(38,66,52,80)(39,79,53,65)(40,64,54,78)(41,77,55,63)(42,62,56,76) );
G=PermutationGroup([[(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),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,85),(82,86),(83,87),(84,88)], [(2,50),(4,52),(6,54),(8,56),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40),(22,42),(24,44),(26,46),(28,48),(57,89),(58,72),(59,91),(60,74),(61,93),(62,76),(63,95),(64,78),(65,97),(66,80),(67,99),(68,82),(69,101),(70,84),(71,103),(73,105),(75,107),(77,109),(79,111),(81,85),(83,87),(86,100),(88,102),(90,104),(92,106),(94,108),(96,110),(98,112)], [(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)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(57,103),(58,104),(59,105),(60,106),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102)], [(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)], [(1,87,15,101),(2,100,16,86),(3,85,17,99),(4,98,18,112),(5,111,19,97),(6,96,20,110),(7,109,21,95),(8,94,22,108),(9,107,23,93),(10,92,24,106),(11,105,25,91),(12,90,26,104),(13,103,27,89),(14,88,28,102),(29,61,43,75),(30,74,44,60),(31,59,45,73),(32,72,46,58),(33,57,47,71),(34,70,48,84),(35,83,49,69),(36,68,50,82),(37,81,51,67),(38,66,52,80),(39,79,53,65),(40,64,54,78),(41,77,55,63),(42,62,56,76)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 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 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4⋊6D14 |
| kernel | C24.30D14 | C23.D14 | D14.D4 | D14⋊D4 | Dic7.D4 | C28.48D4 | C4×C7⋊D4 | C28⋊7D4 | C24⋊D7 | C14×C22⋊C4 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C22 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 12 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of C24.30D14 ►in GL6(𝔽29)
| 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 | 3 | 0 | 1 | 0 |
| 0 | 0 | 13 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 28 | 0 | 0 |
| 0 | 0 | 26 | 0 | 28 | 0 |
| 0 | 0 | 1 | 0 | 20 | 1 |
| 28 | 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 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 2 | 0 | 0 |
| 0 | 0 | 18 | 12 | 0 | 0 |
| 0 | 0 | 24 | 26 | 21 | 5 |
| 0 | 0 | 5 | 16 | 23 | 8 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 16 | 19 |
| 0 | 0 | 2 | 0 | 10 | 27 |
| 0 | 0 | 24 | 28 | 5 | 15 |
| 0 | 0 | 21 | 10 | 12 | 7 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,3,13,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,26,1,0,0,0,28,0,0,0,0,0,0,28,20,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,17,18,24,5,0,0,2,12,26,16,0,0,0,0,21,23,0,0,0,0,5,8],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,17,2,24,21,0,0,0,0,28,10,0,0,16,10,5,12,0,0,19,27,15,7] >;
C24.30D14 in GAP, Magma, Sage, TeX
C_2^4._{30}D_{14} % in TeX
G:=Group("C2^4.30D14"); // GroupNames label
G:=SmallGroup(448,947);
// by ID
G=gap.SmallGroup(448,947);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,100,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,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