metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.24D14, C14.22+ 1+4, C23⋊3(C4×D7), (C22×C4)⋊3D14, C22⋊C4⋊51D14, C14.9(C23×C4), D14⋊C4⋊57C22, (C2×C14).30C24, D14.2(C22×C4), Dic7⋊4D4⋊39C2, C2.1(D4⋊6D14), (C2×C28).571C23, Dic7⋊C4⋊58C22, (C22×C28)⋊34C22, C7⋊1(C22.11C24), (C4×Dic7)⋊46C22, C23.D7⋊67C22, Dic7.3(C22×C4), C22.19(C23×D7), (C23×C14).56C22, (C22×Dic7)⋊5C22, (C23×D7).30C22, C23.220(C22×D7), C23.11D14⋊24C2, (C22×C14).122C23, (C2×Dic7).177C23, (C22×D7).150C23, C7⋊D4⋊9(C2×C4), (C2×C7⋊D4)⋊9C4, (C4×C7⋊D4)⋊33C2, (C2×C22⋊C4)⋊7D7, (C2×C4×D7)⋊39C22, C2.11(D7×C22×C4), C22.24(C2×C4×D7), (D7×C22⋊C4)⋊23C2, (C22×D7)⋊6(C2×C4), (C14×C22⋊C4)⋊26C2, (C22×C14)⋊10(C2×C4), (C2×Dic7)⋊10(C2×C4), (C2×C23.D7)⋊15C2, (C22×C7⋊D4).9C2, (C7×C22⋊C4)⋊61C22, (C2×C14).18(C22×C4), (C2×C4).257(C22×D7), (C2×C7⋊D4).88C22, SmallGroup(448,939)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.24D14
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, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1492 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C22.11C24, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C23.11D14, D7×C22⋊C4, Dic7⋊4D4, C4×C7⋊D4, C2×C23.D7, C14×C22⋊C4, C22×C7⋊D4, C24.24D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C23×C4, 2+ 1+4, C4×D7, C22×D7, C22.11C24, C2×C4×D7, C23×D7, D7×C22×C4, D4⋊6D14, C24.24D14
►On 112 points
Generators in S112
(2 53)(4 55)(6 29)(8 31)(10 33)(12 35)(14 37)(16 39)(18 41)(20 43)(22 45)(24 47)(26 49)(28 51)(58 106)(60 108)(62 110)(64 112)(66 86)(68 88)(70 90)(72 92)(74 94)(76 96)(78 98)(80 100)(82 102)(84 104)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(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)
(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 52)(2 53)(3 54)(4 55)(5 56)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 46 43 32)(30 31 44 45)(33 42 47 56)(34 55 48 41)(35 40 49 54)(36 53 50 39)(37 38 51 52)(57 106 71 92)(58 91 72 105)(59 104 73 90)(60 89 74 103)(61 102 75 88)(62 87 76 101)(63 100 77 86)(64 85 78 99)(65 98 79 112)(66 111 80 97)(67 96 81 110)(68 109 82 95)(69 94 83 108)(70 107 84 93)
G:=sub<Sym(112)| (2,53)(4,55)(6,29)(8,31)(10,33)(12,35)(14,37)(16,39)(18,41)(20,43)(22,45)(24,47)(26,49)(28,51)(58,106)(60,108)(62,110)(64,112)(66,86)(68,88)(70,90)(72,92)(74,94)(76,96)(78,98)(80,100)(82,102)(84,104), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(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), (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), (1,52)(2,53)(3,54)(4,55)(5,56)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,46,43,32)(30,31,44,45)(33,42,47,56)(34,55,48,41)(35,40,49,54)(36,53,50,39)(37,38,51,52)(57,106,71,92)(58,91,72,105)(59,104,73,90)(60,89,74,103)(61,102,75,88)(62,87,76,101)(63,100,77,86)(64,85,78,99)(65,98,79,112)(66,111,80,97)(67,96,81,110)(68,109,82,95)(69,94,83,108)(70,107,84,93)>;
G:=Group( (2,53)(4,55)(6,29)(8,31)(10,33)(12,35)(14,37)(16,39)(18,41)(20,43)(22,45)(24,47)(26,49)(28,51)(58,106)(60,108)(62,110)(64,112)(66,86)(68,88)(70,90)(72,92)(74,94)(76,96)(78,98)(80,100)(82,102)(84,104), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(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), (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), (1,52)(2,53)(3,54)(4,55)(5,56)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,46,43,32)(30,31,44,45)(33,42,47,56)(34,55,48,41)(35,40,49,54)(36,53,50,39)(37,38,51,52)(57,106,71,92)(58,91,72,105)(59,104,73,90)(60,89,74,103)(61,102,75,88)(62,87,76,101)(63,100,77,86)(64,85,78,99)(65,98,79,112)(66,111,80,97)(67,96,81,110)(68,109,82,95)(69,94,83,108)(70,107,84,93) );
G=PermutationGroup([[(2,53),(4,55),(6,29),(8,31),(10,33),(12,35),(14,37),(16,39),(18,41),(20,43),(22,45),(24,47),(26,49),(28,51),(58,106),(60,108),(62,110),(64,112),(66,86),(68,88),(70,90),(72,92),(74,94),(76,96),(78,98),(80,100),(82,102),(84,104)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(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)], [(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)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104)], [(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)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,46,43,32),(30,31,44,45),(33,42,47,56),(34,55,48,41),(35,40,49,54),(36,53,50,39),(37,38,51,52),(57,106,71,92),(58,91,72,105),(59,104,73,90),(60,89,74,103),(61,102,75,88),(62,87,76,101),(63,100,77,86),(64,85,78,99),(65,98,79,112),(66,111,80,97),(67,96,81,110),(68,109,82,95),(69,94,83,108),(70,107,84,93)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | D14 | D14 | D14 | C4×D7 | 2+ 1+4 | D4⋊6D14 |
| kernel | C24.24D14 | C23.11D14 | D7×C22⋊C4 | Dic7⋊4D4 | C4×C7⋊D4 | C2×C23.D7 | C14×C22⋊C4 | C22×C7⋊D4 | C2×C7⋊D4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C14 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 16 | 3 | 12 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of C24.24D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 16 |
| 0 | 0 | 0 | 0 | 13 | 24 |
| 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 |
| 20 | 19 | 0 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 |
| 0 | 0 | 0 | 0 | 8 | 26 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 7 | 13 | 0 | 0 | 0 | 0 |
| 14 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 |
| 0 | 0 | 0 | 0 | 26 | 8 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 3 | 21 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,0,0,0,0,5,13,0,0,0,0,16,24],[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],[20,2,0,0,0,0,19,12,0,0,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,21,8,0,0,0,0,21,26,0,0],[7,14,0,0,0,0,13,22,0,0,0,0,0,0,0,0,8,3,0,0,0,0,8,21,0,0,21,26,0,0,0,0,21,8,0,0] >;
C24.24D14 in GAP, Magma, Sage, TeX
C_2^4._{24}D_{14} % in TeX
G:=Group("C2^4.24D14"); // GroupNames label
G:=SmallGroup(448,939);
// by ID
G=gap.SmallGroup(448,939);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,570,80,18822]);
// 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,b*c=c*b,f*b*f^-1=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