metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.32D14, C14.242+ 1+4, (D4×Dic7)⋊11C2, C22≀C2.2D7, C22⋊C4.1D14, (C2×D4).149D14, (C2×C28).26C23, Dic7⋊C4⋊7C22, C4⋊Dic7⋊24C22, C28.17D4⋊11C2, (C2×C14).131C24, (C4×Dic7)⋊13C22, C23.D7⋊12C22, C2.26(D4⋊6D14), C22⋊Dic14⋊12C2, C7⋊4(C22.45C24), (C2×Dic14)⋊19C22, (D4×C14).110C22, C23.18D14⋊3C2, C23.11D14⋊2C2, C23.D14⋊11C2, (C23×C14).67C22, C23.176(C22×D7), C22.152(C23×D7), C22.17(D4⋊2D7), (C22×C14).180C23, (C2×Dic7).220C23, (C22×Dic7)⋊11C22, C14.76(C2×C4○D4), (C7×C22≀C2).2C2, C2.27(C2×D4⋊2D7), (C2×C23.D7)⋊19C2, (C2×C4).26(C22×D7), (C2×C14).43(C4○D4), (C7×C22⋊C4).2C22, SmallGroup(448,1040)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.32D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 940 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C22.45C24, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C2×Dic14, C22×Dic7, C22×Dic7, D4×C14, D4×C14, C23×C14, C23.11D14, C22⋊Dic14, C23.D14, D4×Dic7, C23.18D14, C23.18D14, C28.17D4, C2×C23.D7, C7×C22≀C2, C24.32D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, D4⋊6D14, C24.32D14
►On 112 points
Generators in S112
(2 104)(4 106)(6 108)(8 110)(10 112)(12 100)(14 102)(15 38)(17 40)(19 42)(21 30)(23 32)(25 34)(27 36)(43 94)(45 96)(47 98)(49 86)(51 88)(53 90)(55 92)(58 79)(60 81)(62 83)(64 71)(66 73)(68 75)(70 77)
(2 21)(4 23)(6 25)(8 27)(10 15)(12 17)(14 19)(30 104)(32 106)(34 108)(36 110)(38 112)(40 100)(42 102)(43 94)(44 63)(45 96)(46 65)(47 98)(48 67)(49 86)(50 69)(51 88)(52 57)(53 90)(54 59)(55 92)(56 61)(58 79)(60 81)(62 83)(64 71)(66 73)(68 75)(70 77)(72 97)(74 85)(76 87)(78 89)(80 91)(82 93)(84 95)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 99)(12 100)(13 101)(14 102)(15 38)(16 39)(17 40)(18 41)(19 42)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(43 94)(44 95)(45 96)(46 97)(47 98)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 15)(11 16)(12 17)(13 18)(14 19)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 99)(40 100)(41 101)(42 102)(43 83)(44 84)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)
(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 81 20 55)(2 80 21 54)(3 79 22 53)(4 78 23 52)(5 77 24 51)(6 76 25 50)(7 75 26 49)(8 74 27 48)(9 73 28 47)(10 72 15 46)(11 71 16 45)(12 84 17 44)(13 83 18 43)(14 82 19 56)(29 92 103 60)(30 91 104 59)(31 90 105 58)(32 89 106 57)(33 88 107 70)(34 87 108 69)(35 86 109 68)(36 85 110 67)(37 98 111 66)(38 97 112 65)(39 96 99 64)(40 95 100 63)(41 94 101 62)(42 93 102 61)
G:=sub<Sym(112)| (2,104)(4,106)(6,108)(8,110)(10,112)(12,100)(14,102)(15,38)(17,40)(19,42)(21,30)(23,32)(25,34)(27,36)(43,94)(45,96)(47,98)(49,86)(51,88)(53,90)(55,92)(58,79)(60,81)(62,83)(64,71)(66,73)(68,75)(70,77), (2,21)(4,23)(6,25)(8,27)(10,15)(12,17)(14,19)(30,104)(32,106)(34,108)(36,110)(38,112)(40,100)(42,102)(43,94)(44,63)(45,96)(46,65)(47,98)(48,67)(49,86)(50,69)(51,88)(52,57)(53,90)(54,59)(55,92)(56,61)(58,79)(60,81)(62,83)(64,71)(66,73)(68,75)(70,77)(72,97)(74,85)(76,87)(78,89)(80,91)(82,93)(84,95), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,99)(12,100)(13,101)(14,102)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,15)(11,16)(12,17)(13,18)(14,19)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,99)(40,100)(41,101)(42,102)(43,83)(44,84)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88), (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), (1,81,20,55)(2,80,21,54)(3,79,22,53)(4,78,23,52)(5,77,24,51)(6,76,25,50)(7,75,26,49)(8,74,27,48)(9,73,28,47)(10,72,15,46)(11,71,16,45)(12,84,17,44)(13,83,18,43)(14,82,19,56)(29,92,103,60)(30,91,104,59)(31,90,105,58)(32,89,106,57)(33,88,107,70)(34,87,108,69)(35,86,109,68)(36,85,110,67)(37,98,111,66)(38,97,112,65)(39,96,99,64)(40,95,100,63)(41,94,101,62)(42,93,102,61)>;
G:=Group( (2,104)(4,106)(6,108)(8,110)(10,112)(12,100)(14,102)(15,38)(17,40)(19,42)(21,30)(23,32)(25,34)(27,36)(43,94)(45,96)(47,98)(49,86)(51,88)(53,90)(55,92)(58,79)(60,81)(62,83)(64,71)(66,73)(68,75)(70,77), (2,21)(4,23)(6,25)(8,27)(10,15)(12,17)(14,19)(30,104)(32,106)(34,108)(36,110)(38,112)(40,100)(42,102)(43,94)(44,63)(45,96)(46,65)(47,98)(48,67)(49,86)(50,69)(51,88)(52,57)(53,90)(54,59)(55,92)(56,61)(58,79)(60,81)(62,83)(64,71)(66,73)(68,75)(70,77)(72,97)(74,85)(76,87)(78,89)(80,91)(82,93)(84,95), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,99)(12,100)(13,101)(14,102)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,15)(11,16)(12,17)(13,18)(14,19)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,99)(40,100)(41,101)(42,102)(43,83)(44,84)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88), (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), (1,81,20,55)(2,80,21,54)(3,79,22,53)(4,78,23,52)(5,77,24,51)(6,76,25,50)(7,75,26,49)(8,74,27,48)(9,73,28,47)(10,72,15,46)(11,71,16,45)(12,84,17,44)(13,83,18,43)(14,82,19,56)(29,92,103,60)(30,91,104,59)(31,90,105,58)(32,89,106,57)(33,88,107,70)(34,87,108,69)(35,86,109,68)(36,85,110,67)(37,98,111,66)(38,97,112,65)(39,96,99,64)(40,95,100,63)(41,94,101,62)(42,93,102,61) );
G=PermutationGroup([[(2,104),(4,106),(6,108),(8,110),(10,112),(12,100),(14,102),(15,38),(17,40),(19,42),(21,30),(23,32),(25,34),(27,36),(43,94),(45,96),(47,98),(49,86),(51,88),(53,90),(55,92),(58,79),(60,81),(62,83),(64,71),(66,73),(68,75),(70,77)], [(2,21),(4,23),(6,25),(8,27),(10,15),(12,17),(14,19),(30,104),(32,106),(34,108),(36,110),(38,112),(40,100),(42,102),(43,94),(44,63),(45,96),(46,65),(47,98),(48,67),(49,86),(50,69),(51,88),(52,57),(53,90),(54,59),(55,92),(56,61),(58,79),(60,81),(62,83),(64,71),(66,73),(68,75),(70,77),(72,97),(74,85),(76,87),(78,89),(80,91),(82,93),(84,95)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,99),(12,100),(13,101),(14,102),(15,38),(16,39),(17,40),(18,41),(19,42),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(43,94),(44,95),(45,96),(46,97),(47,98),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,15),(11,16),(12,17),(13,18),(14,19),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,99),(40,100),(41,101),(42,102),(43,83),(44,84),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,85),(68,86),(69,87),(70,88)], [(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)], [(1,81,20,55),(2,80,21,54),(3,79,22,53),(4,78,23,52),(5,77,24,51),(6,76,25,50),(7,75,26,49),(8,74,27,48),(9,73,28,47),(10,72,15,46),(11,71,16,45),(12,84,17,44),(13,83,18,43),(14,82,19,56),(29,92,103,60),(30,91,104,59),(31,90,105,58),(32,89,106,57),(33,88,107,70),(34,87,108,69),(35,86,109,68),(36,85,110,67),(37,98,111,66),(38,97,112,65),(39,96,99,64),(40,95,100,63),(41,94,101,62),(42,93,102,61)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ 1+4 | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.32D14 | C23.11D14 | C22⋊Dic14 | C23.D14 | D4×Dic7 | C23.18D14 | C28.17D4 | C2×C23.D7 | C7×C22≀C2 | C22≀C2 | C2×C14 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 3 | 8 | 9 | 9 | 3 | 1 | 12 | 6 |
Matrix representation of C24.32D14 ►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 | 6 | 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 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 25 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,25,0,0,0,0,4,18,0,0,0,0,0,0,28,0,0,0,0,0,10,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,3,0,0,0,0,7,26,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;
C24.32D14 in GAP, Magma, Sage, TeX
C_2^4._{32}D_{14} % in TeX
G:=Group("C2^4.32D14"); // GroupNames label
G:=SmallGroup(448,1040);
// by ID
G=gap.SmallGroup(448,1040);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,1571,297,18822]);
// 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,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=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