metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.2C4≀C2, C4⋊Dic7⋊2C4, (C2×Dic14)⋊1C4, (C2×C4).104D28, (C2×C28).222D4, (C2×C14).13Q16, C14.2(C23⋊C4), (C2×C14).28SD16, (C22×C4).53D14, C2.5(Dic14⋊C4), C14.1(Q8⋊C4), C28.55D4.1C2, C28.48D4.7C2, C2.3(C14.Q16), (C22×C14).175D4, C7⋊1(C23.31D4), C23.71(C7⋊D4), C22.4(D4.D7), C22.4(C7⋊Q16), C2.C42.6D7, C22.55(D14⋊C4), (C22×C28).90C22, C2.5(C23.1D14), (C2×C4).9(C4×D7), (C2×C28).21(C2×C4), (C2×C14).34(C22⋊C4), (C7×C2.C42).13C2, SmallGroup(448,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic7⋊C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, cac-1=a-1, dad-1=ab7, cbc-1=b-1, bd=db, dcd-1=a-1b7c >
Subgroups: 348 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C22⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.31D4, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×Dic14, C22×C28, C22×C28, C28.55D4, C7×C2.C42, C28.48D4, C4⋊Dic7⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, SD16, Q16, D14, C23⋊C4, Q8⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, C23.31D4, D14⋊C4, D4.D7, C7⋊Q16, Dic14⋊C4, C23.1D14, C14.Q16, C4⋊Dic7⋊C4
►On 112 points
Generators in S112
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 95 77 105)(58 96 78 106)(59 97 79 107)(60 98 80 108)(61 85 81 109)(62 86 82 110)(63 87 83 111)(64 88 84 112)(65 89 71 99)(66 90 72 100)(67 91 73 101)(68 92 74 102)(69 93 75 103)(70 94 76 104)
(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 68 8 61)(2 67 9 60)(3 66 10 59)(4 65 11 58)(5 64 12 57)(6 63 13 70)(7 62 14 69)(15 74 22 81)(16 73 23 80)(17 72 24 79)(18 71 25 78)(19 84 26 77)(20 83 27 76)(21 82 28 75)(29 92 36 85)(30 91 37 98)(31 90 38 97)(32 89 39 96)(33 88 40 95)(34 87 41 94)(35 86 42 93)(43 102 50 109)(44 101 51 108)(45 100 52 107)(46 99 53 106)(47 112 54 105)(48 111 55 104)(49 110 56 103)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(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 88 84 105)(58 89 71 106)(59 90 72 107)(60 91 73 108)(61 92 74 109)(62 93 75 110)(63 94 76 111)(64 95 77 112)(65 96 78 99)(66 97 79 100)(67 98 80 101)(68 85 81 102)(69 86 82 103)(70 87 83 104)
G:=sub<Sym(112)| (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,95,77,105)(58,96,78,106)(59,97,79,107)(60,98,80,108)(61,85,81,109)(62,86,82,110)(63,87,83,111)(64,88,84,112)(65,89,71,99)(66,90,72,100)(67,91,73,101)(68,92,74,102)(69,93,75,103)(70,94,76,104), (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), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,92,36,85)(30,91,37,98)(31,90,38,97)(32,89,39,96)(33,88,40,95)(34,87,41,94)(35,86,42,93)(43,102,50,109)(44,101,51,108)(45,100,52,107)(46,99,53,106)(47,112,54,105)(48,111,55,104)(49,110,56,103), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(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)(57,88,84,105)(58,89,71,106)(59,90,72,107)(60,91,73,108)(61,92,74,109)(62,93,75,110)(63,94,76,111)(64,95,77,112)(65,96,78,99)(66,97,79,100)(67,98,80,101)(68,85,81,102)(69,86,82,103)(70,87,83,104)>;
G:=Group( (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,95,77,105)(58,96,78,106)(59,97,79,107)(60,98,80,108)(61,85,81,109)(62,86,82,110)(63,87,83,111)(64,88,84,112)(65,89,71,99)(66,90,72,100)(67,91,73,101)(68,92,74,102)(69,93,75,103)(70,94,76,104), (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), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,92,36,85)(30,91,37,98)(31,90,38,97)(32,89,39,96)(33,88,40,95)(34,87,41,94)(35,86,42,93)(43,102,50,109)(44,101,51,108)(45,100,52,107)(46,99,53,106)(47,112,54,105)(48,111,55,104)(49,110,56,103), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(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)(57,88,84,105)(58,89,71,106)(59,90,72,107)(60,91,73,108)(61,92,74,109)(62,93,75,110)(63,94,76,111)(64,95,77,112)(65,96,78,99)(66,97,79,100)(67,98,80,101)(68,85,81,102)(69,86,82,103)(70,87,83,104) );
G=PermutationGroup([[(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,95,77,105),(58,96,78,106),(59,97,79,107),(60,98,80,108),(61,85,81,109),(62,86,82,110),(63,87,83,111),(64,88,84,112),(65,89,71,99),(66,90,72,100),(67,91,73,101),(68,92,74,102),(69,93,75,103),(70,94,76,104)], [(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)], [(1,68,8,61),(2,67,9,60),(3,66,10,59),(4,65,11,58),(5,64,12,57),(6,63,13,70),(7,62,14,69),(15,74,22,81),(16,73,23,80),(17,72,24,79),(18,71,25,78),(19,84,26,77),(20,83,27,76),(21,82,28,75),(29,92,36,85),(30,91,37,98),(31,90,38,97),(32,89,39,96),(33,88,40,95),(34,87,41,94),(35,86,42,93),(43,102,50,109),(44,101,51,108),(45,100,52,107),(46,99,53,106),(47,112,54,105),(48,111,55,104),(49,110,56,103)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(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),(57,88,84,105),(58,89,71,106),(59,90,72,107),(60,91,73,108),(61,92,74,109),(62,93,75,110),(63,94,76,111),(64,95,77,112),(65,96,78,99),(66,97,79,100),(67,98,80,101),(68,85,81,102),(69,86,82,103),(70,87,83,104)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | SD16 | Q16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | Dic14⋊C4 | C23⋊C4 | D4.D7 | C7⋊Q16 | C23.1D14 |
| kernel | C4⋊Dic7⋊C4 | C28.55D4 | C7×C2.C42 | C28.48D4 | C4⋊Dic7 | C2×Dic14 | C2×C28 | C22×C14 | C2.C42 | C2×C14 | C2×C14 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 | C14 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C4⋊Dic7⋊C4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 17 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 22 | 109 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 19 | 34 | 0 | 0 | 0 | 0 |
| 16 | 94 | 0 | 0 | 0 | 0 |
| 0 | 0 | 110 | 8 | 0 | 0 |
| 0 | 0 | 112 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 107 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,17,0,0,0,0,0,15,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[28,22,0,0,0,0,0,109,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[19,16,0,0,0,0,34,94,0,0,0,0,0,0,110,112,0,0,0,0,8,3,0,0,0,0,0,0,13,100,0,0,0,0,100,100],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,107,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,112] >;
C4⋊Dic7⋊C4 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_7\rtimes C_4 % in TeX
G:=Group("C4:Dic7:C4"); // GroupNames label
G:=SmallGroup(448,9);
// by ID
G=gap.SmallGroup(448,9);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,1571,570,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^7,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^-1*b^7*c>;
// generators/relations