metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.27D14, C14.32+ 1+4, C7⋊D4⋊8D4, C7⋊1(D4⋊5D4), (C22×C4)⋊8D14, D14⋊D4⋊1C2, C22⋊D28⋊2C2, C28⋊7D4⋊17C2, C22⋊C4⋊40D14, D14.12(C2×D4), (C2×D28)⋊2C22, C24⋊D7⋊2C2, C4⋊Dic7⋊4C22, C22.18(D4×D7), D14⋊C4⋊46C22, D14.D4⋊1C2, C22⋊4(C4○D28), (C2×C14).34C24, Dic7.14(C2×D4), C14.37(C22×D4), Dic7⋊4D4⋊40C2, C23.D7⋊8C22, C2.7(D4⋊6D14), (C2×C28).128C23, Dic7⋊C4⋊49C22, (C22×C28)⋊14C22, Dic7.D4⋊1C2, C22⋊Dic14⋊2C2, (C4×Dic7)⋊47C22, C22.73(C23×D7), (C2×Dic14)⋊48C22, C23.23D14⋊9C2, (C23×C14).60C22, (C23×D7).31C22, C23.221(C22×D7), (C22×C14).387C23, (C2×Dic7).180C23, (C22×D7).152C23, (C22×Dic7).78C22, C2.11(C2×D4×D7), (C4×C7⋊D4)⋊1C2, (C2×C4○D28)⋊3C2, (C2×C4×D7)⋊40C22, (C2×C14)⋊8(C4○D4), (D7×C22⋊C4)⋊24C2, (C2×C22⋊C4)⋊13D7, C2.16(C2×C4○D28), C14.14(C2×C4○D4), (C22×C7⋊D4)⋊5C2, (C2×C7⋊D4)⋊1C22, (C14×C22⋊C4)⋊18C2, (C2×C14).383(C2×D4), (C7×C22⋊C4)⋊53C22, (C2×C4).259(C22×D7), SmallGroup(448,943)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.27D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1716 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C22⋊Dic14, D7×C22⋊C4, Dic7⋊4D4, C22⋊D28, D14.D4, D14⋊D4, Dic7.D4, C4×C7⋊D4, C23.23D14, C28⋊7D4, C24⋊D7, C14×C22⋊C4, C2×C4○D28, C22×C7⋊D4, C24.27D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, C4○D28, D4×D7, C23×D7, C2×C4○D28, C2×D4×D7, D4⋊6D14, C24.27D14
►On 112 points
Generators in S112
(2 101)(4 103)(6 105)(8 107)(10 109)(12 111)(14 85)(16 87)(18 89)(20 91)(22 93)(24 95)(26 97)(28 99)(29 77)(31 79)(33 81)(35 83)(37 57)(39 59)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 57)(27 58)(28 59)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 85)(54 86)(55 87)(56 88)
(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)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
(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 85 15 99)(2 98 16 112)(3 111 17 97)(4 96 18 110)(5 109 19 95)(6 94 20 108)(7 107 21 93)(8 92 22 106)(9 105 23 91)(10 90 24 104)(11 103 25 89)(12 88 26 102)(13 101 27 87)(14 86 28 100)(29 84 43 70)(30 69 44 83)(31 82 45 68)(32 67 46 81)(33 80 47 66)(34 65 48 79)(35 78 49 64)(36 63 50 77)(37 76 51 62)(38 61 52 75)(39 74 53 60)(40 59 54 73)(41 72 55 58)(42 57 56 71)
G:=sub<Sym(112)| (2,101)(4,103)(6,105)(8,107)(10,109)(12,111)(14,85)(16,87)(18,89)(20,91)(22,93)(24,95)(26,97)(28,99)(29,77)(31,79)(33,81)(35,83)(37,57)(39,59)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,85)(54,86)(55,87)(56,88), (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), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,85,15,99)(2,98,16,112)(3,111,17,97)(4,96,18,110)(5,109,19,95)(6,94,20,108)(7,107,21,93)(8,92,22,106)(9,105,23,91)(10,90,24,104)(11,103,25,89)(12,88,26,102)(13,101,27,87)(14,86,28,100)(29,84,43,70)(30,69,44,83)(31,82,45,68)(32,67,46,81)(33,80,47,66)(34,65,48,79)(35,78,49,64)(36,63,50,77)(37,76,51,62)(38,61,52,75)(39,74,53,60)(40,59,54,73)(41,72,55,58)(42,57,56,71)>;
G:=Group( (2,101)(4,103)(6,105)(8,107)(10,109)(12,111)(14,85)(16,87)(18,89)(20,91)(22,93)(24,95)(26,97)(28,99)(29,77)(31,79)(33,81)(35,83)(37,57)(39,59)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,85)(54,86)(55,87)(56,88), (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), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,85,15,99)(2,98,16,112)(3,111,17,97)(4,96,18,110)(5,109,19,95)(6,94,20,108)(7,107,21,93)(8,92,22,106)(9,105,23,91)(10,90,24,104)(11,103,25,89)(12,88,26,102)(13,101,27,87)(14,86,28,100)(29,84,43,70)(30,69,44,83)(31,82,45,68)(32,67,46,81)(33,80,47,66)(34,65,48,79)(35,78,49,64)(36,63,50,77)(37,76,51,62)(38,61,52,75)(39,74,53,60)(40,59,54,73)(41,72,55,58)(42,57,56,71) );
G=PermutationGroup([[(2,101),(4,103),(6,105),(8,107),(10,109),(12,111),(14,85),(16,87),(18,89),(20,91),(22,93),(24,95),(26,97),(28,99),(29,77),(31,79),(33,81),(35,83),(37,57),(39,59),(41,61),(43,63),(45,65),(47,67),(49,69),(51,71),(53,73),(55,75)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,57),(27,58),(28,59),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,85),(54,86),(55,87),(56,88)], [(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)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)], [(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)], [(1,85,15,99),(2,98,16,112),(3,111,17,97),(4,96,18,110),(5,109,19,95),(6,94,20,108),(7,107,21,93),(8,92,22,106),(9,105,23,91),(10,90,24,104),(11,103,25,89),(12,88,26,102),(13,101,27,87),(14,86,28,100),(29,84,43,70),(30,69,44,83),(31,82,45,68),(32,67,46,81),(33,80,47,66),(34,65,48,79),(35,78,49,64),(36,63,50,77),(37,76,51,62),(38,61,52,75),(39,74,53,60),(40,59,54,73),(41,72,55,58),(42,57,56,71)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 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 | 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 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4×D7 | D4⋊6D14 |
| kernel | C24.27D14 | C22⋊Dic14 | D7×C22⋊C4 | Dic7⋊4D4 | C22⋊D28 | D14.D4 | D14⋊D4 | Dic7.D4 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | C24⋊D7 | C14×C22⋊C4 | C2×C4○D28 | C22×C7⋊D4 | C7⋊D4 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C22 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 12 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of C24.27D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 25 | 28 |
| 11 | 14 | 0 | 0 |
| 8 | 18 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 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 |
| 17 | 4 | 0 | 0 |
| 23 | 19 | 0 | 0 |
| 0 | 0 | 3 | 16 |
| 0 | 0 | 14 | 26 |
| 19 | 17 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 26 | 13 |
| 0 | 0 | 15 | 3 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,25,0,0,0,28],[11,8,0,0,14,18,0,0,0,0,28,0,0,0,0,28],[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],[17,23,0,0,4,19,0,0,0,0,3,14,0,0,16,26],[19,6,0,0,17,10,0,0,0,0,26,15,0,0,13,3] >;
C24.27D14 in GAP, Magma, Sage, TeX
C_2^4._{27}D_{14} % in TeX
G:=Group("C2^4.27D14"); // GroupNames label
G:=SmallGroup(448,943);
// by ID
G=gap.SmallGroup(448,943);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,675,18822]);
// 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,e*a*e^-1=f*a*f^-1=a*d=d*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