metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊11D14, C14.152+ 1+4, D4⋊9(C4×D7), (C4×D4)⋊7D7, (D4×D7)⋊5C4, C4⋊C4⋊56D14, (D4×C28)⋊9C2, (C4×D28)⋊24C2, D28⋊13(C2×C4), (D4×Dic7)⋊8C2, (C22×C4)⋊4D14, (C4×C28)⋊16C22, C22⋊C4⋊53D14, D28⋊C4⋊14C2, D14⋊C4⋊62C22, (C2×D4).245D14, C42⋊D7⋊11C2, C14.22(C23×C4), (C2×C14).89C24, C28.32(C22×C4), C4⋊Dic7⋊73C22, D14.8(C22×C4), Dic7⋊4D4⋊45C2, C2.3(D4⋊8D14), C2.3(D4⋊6D14), (C2×C28).587C23, Dic7⋊C4⋊64C22, (C22×C28)⋊36C22, C7⋊3(C22.11C24), (C4×Dic7)⋊11C22, C23.D7⋊48C22, Dic7.9(C22×C4), C22.32(C23×D7), (D4×C14).253C22, (C2×D28).258C22, (C22×Dic7)⋊8C22, (C23×D7).38C22, C23.168(C22×D7), (C22×C14).159C23, (C2×Dic7).201C23, (C22×D7).168C23, (C2×D4×D7).7C2, C4.32(C2×C4×D7), (C4×D7)⋊3(C2×C4), C7⋊D4⋊3(C2×C4), (C7×D4)⋊12(C2×C4), (C4×C7⋊D4)⋊40C2, C22.2(C2×C4×D7), (C2×C4×D7)⋊46C22, (C2×D14⋊C4)⋊34C2, C4⋊C4⋊7D7⋊14C2, C2.24(D7×C22×C4), (C7×C4⋊C4)⋊56C22, (D7×C22⋊C4)⋊27C2, (C22×D7)⋊8(C2×C4), (C2×C14).2(C22×C4), (C7×C22⋊C4)⋊63C22, (C2×C4).282(C22×D7), (C2×C7⋊D4).110C22, SmallGroup(448,998)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊11D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1684 in 338 conjugacy classes, 151 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D4, C22×D4, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22.11C24, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C42⋊D7, C4×D28, D7×C22⋊C4, Dic7⋊4D4, C4⋊C4⋊7D7, D28⋊C4, C2×D14⋊C4, C4×C7⋊D4, D4×Dic7, D4×C28, C2×D4×D7, C42⋊11D14
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, D4⋊8D14, C42⋊11D14
►On 112 points
(1 8 43 50)(2 51 44 9)(3 10 45 52)(4 53 46 11)(5 12 47 54)(6 55 48 13)(7 14 49 56)(15 22 86 93)(16 94 87 23)(17 24 88 95)(18 96 89 25)(19 26 90 97)(20 98 91 27)(21 28 92 85)(29 36 72 79)(30 80 73 37)(31 38 74 81)(32 82 75 39)(33 40 76 83)(34 84 77 41)(35 42 78 71)(57 110 103 64)(58 65 104 111)(59 112 105 66)(60 67 106 99)(61 100 107 68)(62 69 108 101)(63 102 109 70)
(1 58 29 90)(2 59 30 91)(3 60 31 92)(4 61 32 93)(5 62 33 94)(6 63 34 95)(7 64 35 96)(8 65 36 97)(9 66 37 98)(10 67 38 85)(11 68 39 86)(12 69 40 87)(13 70 41 88)(14 57 42 89)(15 53 100 82)(16 54 101 83)(17 55 102 84)(18 56 103 71)(19 43 104 72)(20 44 105 73)(21 45 106 74)(22 46 107 75)(23 47 108 76)(24 48 109 77)(25 49 110 78)(26 50 111 79)(27 51 112 80)(28 52 99 81)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 61)(16 60)(17 59)(18 58)(19 57)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(43 71)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 112)(96 111)(97 110)(98 109)
G:=sub<Sym(112)| (1,8,43,50)(2,51,44,9)(3,10,45,52)(4,53,46,11)(5,12,47,54)(6,55,48,13)(7,14,49,56)(15,22,86,93)(16,94,87,23)(17,24,88,95)(18,96,89,25)(19,26,90,97)(20,98,91,27)(21,28,92,85)(29,36,72,79)(30,80,73,37)(31,38,74,81)(32,82,75,39)(33,40,76,83)(34,84,77,41)(35,42,78,71)(57,110,103,64)(58,65,104,111)(59,112,105,66)(60,67,106,99)(61,100,107,68)(62,69,108,101)(63,102,109,70), (1,58,29,90)(2,59,30,91)(3,60,31,92)(4,61,32,93)(5,62,33,94)(6,63,34,95)(7,64,35,96)(8,65,36,97)(9,66,37,98)(10,67,38,85)(11,68,39,86)(12,69,40,87)(13,70,41,88)(14,57,42,89)(15,53,100,82)(16,54,101,83)(17,55,102,84)(18,56,103,71)(19,43,104,72)(20,44,105,73)(21,45,106,74)(22,46,107,75)(23,47,108,76)(24,48,109,77)(25,49,110,78)(26,50,111,79)(27,51,112,80)(28,52,99,81), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,61)(16,60)(17,59)(18,58)(19,57)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(43,71)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,112)(96,111)(97,110)(98,109)>;
G:=Group( (1,8,43,50)(2,51,44,9)(3,10,45,52)(4,53,46,11)(5,12,47,54)(6,55,48,13)(7,14,49,56)(15,22,86,93)(16,94,87,23)(17,24,88,95)(18,96,89,25)(19,26,90,97)(20,98,91,27)(21,28,92,85)(29,36,72,79)(30,80,73,37)(31,38,74,81)(32,82,75,39)(33,40,76,83)(34,84,77,41)(35,42,78,71)(57,110,103,64)(58,65,104,111)(59,112,105,66)(60,67,106,99)(61,100,107,68)(62,69,108,101)(63,102,109,70), (1,58,29,90)(2,59,30,91)(3,60,31,92)(4,61,32,93)(5,62,33,94)(6,63,34,95)(7,64,35,96)(8,65,36,97)(9,66,37,98)(10,67,38,85)(11,68,39,86)(12,69,40,87)(13,70,41,88)(14,57,42,89)(15,53,100,82)(16,54,101,83)(17,55,102,84)(18,56,103,71)(19,43,104,72)(20,44,105,73)(21,45,106,74)(22,46,107,75)(23,47,108,76)(24,48,109,77)(25,49,110,78)(26,50,111,79)(27,51,112,80)(28,52,99,81), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,61)(16,60)(17,59)(18,58)(19,57)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(43,71)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,112)(96,111)(97,110)(98,109) );
G=PermutationGroup([[(1,8,43,50),(2,51,44,9),(3,10,45,52),(4,53,46,11),(5,12,47,54),(6,55,48,13),(7,14,49,56),(15,22,86,93),(16,94,87,23),(17,24,88,95),(18,96,89,25),(19,26,90,97),(20,98,91,27),(21,28,92,85),(29,36,72,79),(30,80,73,37),(31,38,74,81),(32,82,75,39),(33,40,76,83),(34,84,77,41),(35,42,78,71),(57,110,103,64),(58,65,104,111),(59,112,105,66),(60,67,106,99),(61,100,107,68),(62,69,108,101),(63,102,109,70)], [(1,58,29,90),(2,59,30,91),(3,60,31,92),(4,61,32,93),(5,62,33,94),(6,63,34,95),(7,64,35,96),(8,65,36,97),(9,66,37,98),(10,67,38,85),(11,68,39,86),(12,69,40,87),(13,70,41,88),(14,57,42,89),(15,53,100,82),(16,54,101,83),(17,55,102,84),(18,56,103,71),(19,43,104,72),(20,44,105,73),(21,45,106,74),(22,46,107,75),(23,47,108,76),(24,48,109,77),(25,49,110,78),(26,50,111,79),(27,51,112,80),(28,52,99,81)], [(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,61),(16,60),(17,59),(18,58),(19,57),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(43,71),(44,84),(45,83),(46,82),(47,81),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,112),(96,111),(97,110),(98,109)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2M | 4A | ··· | 4J | 4K | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 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 | C4 | D7 | D14 | D14 | D14 | D14 | D14 | C4×D7 | 2+ 1+4 | D4⋊6D14 | D4⋊8D14 |
| kernel | C42⋊11D14 | C42⋊D7 | C4×D28 | D7×C22⋊C4 | Dic7⋊4D4 | C4⋊C4⋊7D7 | D28⋊C4 | C2×D14⋊C4 | C4×C7⋊D4 | D4×Dic7 | D4×C28 | C2×D4×D7 | D4×D7 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 16 | 3 | 3 | 6 | 3 | 6 | 3 | 24 | 2 | 6 | 6 |
Matrix representation of C42⋊11D14 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 19 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 19 | 0 | 0 | 0 | 0 |
| 24 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,3,0,0,0,0,19,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[3,24,0,0,0,0,19,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,28,0,0] >;
C42⋊11D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}D_{14} % in TeX
G:=Group("C4^2:11D14"); // GroupNames label
G:=SmallGroup(448,998);
// by ID
G=gap.SmallGroup(448,998);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations