metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D4×C14)⋊6C4, (C2×D4)⋊4Dic7, C28.203(C2×D4), (C2×C28).189D4, D4.5(C2×Dic7), (C22×D4).2D7, (C2×D4).196D14, D4⋊Dic7⋊37C2, C28.80(C22×C4), C4⋊Dic7⋊68C22, C4.8(C23.D7), C28.31(C22⋊C4), (C2×C28).470C23, (C22×C14).195D4, (C22×C4).147D14, C23.84(C7⋊D4), C7⋊4(C23.37D4), C4.10(C22×Dic7), C14.101(C8⋊C22), C2.5(D4.D14), (D4×C14).238C22, C23.21D14⋊18C2, (C22×C28).195C22, C22.20(C23.D7), (C2×C7⋊C8)⋊9C22, (D4×C2×C14).2C2, C4.89(C2×C7⋊D4), (C7×D4).22(C2×C4), (C2×C28).116(C2×C4), C2.8(C2×C23.D7), (C2×C14).552(C2×D4), C14.72(C2×C22⋊C4), (C2×C4.Dic7)⋊17C2, (C2×C4).23(C2×Dic7), C22.90(C2×C7⋊D4), (C2×C4).196(C7⋊D4), (C2×C4).557(C22×D7), (C2×C14).111(C22⋊C4), SmallGroup(448,749)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×C14)⋊6C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 596 in 190 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C23.37D4, C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, D4×C14, D4×C14, C23×C14, D4⋊Dic7, C2×C4.Dic7, C23.21D14, D4×C2×C14, (D4×C14)⋊6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C8⋊C22, C2×Dic7, C7⋊D4, C22×D7, C23.37D4, C23.D7, C22×Dic7, C2×C7⋊D4, D4.D14, C2×C23.D7, (D4×C14)⋊6C4
►On 112 points
Generators in S112
(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 27 12 16)(2 28 13 17)(3 22 14 18)(4 23 8 19)(5 24 9 20)(6 25 10 21)(7 26 11 15)(29 45 40 56)(30 46 41 50)(31 47 42 51)(32 48 36 52)(33 49 37 53)(34 43 38 54)(35 44 39 55)(57 74 64 81)(58 75 65 82)(59 76 66 83)(60 77 67 84)(61 78 68 71)(62 79 69 72)(63 80 70 73)(85 101 92 108)(86 102 93 109)(87 103 94 110)(88 104 95 111)(89 105 96 112)(90 106 97 99)(91 107 98 100)
(1 32)(2 33)(3 34)(4 35)(5 29)(6 30)(7 31)(8 39)(9 40)(10 41)(11 42)(12 36)(13 37)(14 38)(15 47)(16 48)(17 49)(18 43)(19 44)(20 45)(21 46)(22 54)(23 55)(24 56)(25 50)(26 51)(27 52)(28 53)(57 112)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(1 83 48 91)(2 75 49 97)(3 81 43 89)(4 73 44 95)(5 79 45 87)(6 71 46 93)(7 77 47 85)(8 80 55 88)(9 72 56 94)(10 78 50 86)(11 84 51 92)(12 76 52 98)(13 82 53 90)(14 74 54 96)(15 67 31 101)(16 59 32 107)(17 65 33 99)(18 57 34 105)(19 63 35 111)(20 69 29 103)(21 61 30 109)(22 64 38 112)(23 70 39 104)(24 62 40 110)(25 68 41 102)(26 60 42 108)(27 66 36 100)(28 58 37 106)
G:=sub<Sym(112)| (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), (1,27,12,16)(2,28,13,17)(3,22,14,18)(4,23,8,19)(5,24,9,20)(6,25,10,21)(7,26,11,15)(29,45,40,56)(30,46,41,50)(31,47,42,51)(32,48,36,52)(33,49,37,53)(34,43,38,54)(35,44,39,55)(57,74,64,81)(58,75,65,82)(59,76,66,83)(60,77,67,84)(61,78,68,71)(62,79,69,72)(63,80,70,73)(85,101,92,108)(86,102,93,109)(87,103,94,110)(88,104,95,111)(89,105,96,112)(90,106,97,99)(91,107,98,100), (1,32)(2,33)(3,34)(4,35)(5,29)(6,30)(7,31)(8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,47)(16,48)(17,49)(18,43)(19,44)(20,45)(21,46)(22,54)(23,55)(24,56)(25,50)(26,51)(27,52)(28,53)(57,112)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,83,48,91)(2,75,49,97)(3,81,43,89)(4,73,44,95)(5,79,45,87)(6,71,46,93)(7,77,47,85)(8,80,55,88)(9,72,56,94)(10,78,50,86)(11,84,51,92)(12,76,52,98)(13,82,53,90)(14,74,54,96)(15,67,31,101)(16,59,32,107)(17,65,33,99)(18,57,34,105)(19,63,35,111)(20,69,29,103)(21,61,30,109)(22,64,38,112)(23,70,39,104)(24,62,40,110)(25,68,41,102)(26,60,42,108)(27,66,36,100)(28,58,37,106)>;
G:=Group( (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), (1,27,12,16)(2,28,13,17)(3,22,14,18)(4,23,8,19)(5,24,9,20)(6,25,10,21)(7,26,11,15)(29,45,40,56)(30,46,41,50)(31,47,42,51)(32,48,36,52)(33,49,37,53)(34,43,38,54)(35,44,39,55)(57,74,64,81)(58,75,65,82)(59,76,66,83)(60,77,67,84)(61,78,68,71)(62,79,69,72)(63,80,70,73)(85,101,92,108)(86,102,93,109)(87,103,94,110)(88,104,95,111)(89,105,96,112)(90,106,97,99)(91,107,98,100), (1,32)(2,33)(3,34)(4,35)(5,29)(6,30)(7,31)(8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,47)(16,48)(17,49)(18,43)(19,44)(20,45)(21,46)(22,54)(23,55)(24,56)(25,50)(26,51)(27,52)(28,53)(57,112)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,83,48,91)(2,75,49,97)(3,81,43,89)(4,73,44,95)(5,79,45,87)(6,71,46,93)(7,77,47,85)(8,80,55,88)(9,72,56,94)(10,78,50,86)(11,84,51,92)(12,76,52,98)(13,82,53,90)(14,74,54,96)(15,67,31,101)(16,59,32,107)(17,65,33,99)(18,57,34,105)(19,63,35,111)(20,69,29,103)(21,61,30,109)(22,64,38,112)(23,70,39,104)(24,62,40,110)(25,68,41,102)(26,60,42,108)(27,66,36,100)(28,58,37,106) );
G=PermutationGroup([[(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)], [(1,27,12,16),(2,28,13,17),(3,22,14,18),(4,23,8,19),(5,24,9,20),(6,25,10,21),(7,26,11,15),(29,45,40,56),(30,46,41,50),(31,47,42,51),(32,48,36,52),(33,49,37,53),(34,43,38,54),(35,44,39,55),(57,74,64,81),(58,75,65,82),(59,76,66,83),(60,77,67,84),(61,78,68,71),(62,79,69,72),(63,80,70,73),(85,101,92,108),(86,102,93,109),(87,103,94,110),(88,104,95,111),(89,105,96,112),(90,106,97,99),(91,107,98,100)], [(1,32),(2,33),(3,34),(4,35),(5,29),(6,30),(7,31),(8,39),(9,40),(10,41),(11,42),(12,36),(13,37),(14,38),(15,47),(16,48),(17,49),(18,43),(19,44),(20,45),(21,46),(22,54),(23,55),(24,56),(25,50),(26,51),(27,52),(28,53),(57,112),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92)], [(1,83,48,91),(2,75,49,97),(3,81,43,89),(4,73,44,95),(5,79,45,87),(6,71,46,93),(7,77,47,85),(8,80,55,88),(9,72,56,94),(10,78,50,86),(11,84,51,92),(12,76,52,98),(13,82,53,90),(14,74,54,96),(15,67,31,101),(16,59,32,107),(17,65,33,99),(18,57,34,105),(19,63,35,111),(20,69,29,103),(21,61,30,109),(22,64,38,112),(23,70,39,104),(24,62,40,110),(25,68,41,102),(26,60,42,108),(27,66,36,100),(28,58,37,106)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | Dic7 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4.D14 |
| kernel | (D4×C14)⋊6C4 | D4⋊Dic7 | C2×C4.Dic7 | C23.21D14 | D4×C2×C14 | D4×C14 | C2×C28 | C22×C14 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 3 | 12 | 6 | 18 | 6 | 2 | 12 |
Matrix representation of (D4×C14)⋊6C4 ►in GL6(𝔽113)
| 33 | 103 | 0 | 0 | 0 | 0 |
| 42 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 67 | 28 | 0 | 0 | 0 | 0 |
| 106 | 46 | 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 |
G:=sub<GL(6,GF(113))| [33,42,0,0,0,0,103,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,112,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[67,106,0,0,0,0,28,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
(D4×C14)⋊6C4 in GAP, Magma, Sage, TeX
(D_4\times C_{14})\rtimes_6C_4 % in TeX
G:=Group("(D4xC14):6C4"); // GroupNames label
G:=SmallGroup(448,749);
// by ID
G=gap.SmallGroup(448,749);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,387,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations