direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic7, C14⋊2M4(2), C28.41C23, C23.3Dic7, C7⋊C8⋊12C22, (C2×C28).9C4, C7⋊3(C2×M4(2)), C28.36(C2×C4), C4.9(C2×Dic7), (C2×C4).6Dic7, (C22×C4).4D7, (C2×C4).100D14, (C22×C14).7C4, (C22×C28).9C2, C4.41(C22×D7), C14.21(C22×C4), C2.3(C22×Dic7), (C2×C28).100C22, C22.12(C2×Dic7), (C2×C7⋊C8)⋊12C2, (C2×C14).32(C2×C4), SmallGroup(224,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic7
G = < a,b,c,d | a2=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >
Subgroups: 142 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C22×C14, C2×C7⋊C8, C4.Dic7, C22×C28, C2×C4.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, Dic7, D14, C2×M4(2), C2×Dic7, C22×D7, C4.Dic7, C22×Dic7, C2×C4.Dic7
►On 112 points
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 111)(58 112)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)
(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 68 22 61 15 82 8 75)(2 81 23 74 16 67 9 60)(3 66 24 59 17 80 10 73)(4 79 25 72 18 65 11 58)(5 64 26 57 19 78 12 71)(6 77 27 70 20 63 13 84)(7 62 28 83 21 76 14 69)(29 101 50 94 43 87 36 108)(30 86 51 107 44 100 37 93)(31 99 52 92 45 85 38 106)(32 112 53 105 46 98 39 91)(33 97 54 90 47 111 40 104)(34 110 55 103 48 96 41 89)(35 95 56 88 49 109 42 102)
G:=sub<Sym(112)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,111)(58,112)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110), (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112), (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), (1,68,22,61,15,82,8,75)(2,81,23,74,16,67,9,60)(3,66,24,59,17,80,10,73)(4,79,25,72,18,65,11,58)(5,64,26,57,19,78,12,71)(6,77,27,70,20,63,13,84)(7,62,28,83,21,76,14,69)(29,101,50,94,43,87,36,108)(30,86,51,107,44,100,37,93)(31,99,52,92,45,85,38,106)(32,112,53,105,46,98,39,91)(33,97,54,90,47,111,40,104)(34,110,55,103,48,96,41,89)(35,95,56,88,49,109,42,102)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,111)(58,112)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110), (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112), (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), (1,68,22,61,15,82,8,75)(2,81,23,74,16,67,9,60)(3,66,24,59,17,80,10,73)(4,79,25,72,18,65,11,58)(5,64,26,57,19,78,12,71)(6,77,27,70,20,63,13,84)(7,62,28,83,21,76,14,69)(29,101,50,94,43,87,36,108)(30,86,51,107,44,100,37,93)(31,99,52,92,45,85,38,106)(32,112,53,105,46,98,39,91)(33,97,54,90,47,111,40,104)(34,110,55,103,48,96,41,89)(35,95,56,88,49,109,42,102) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,111),(58,112),(59,85),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,109),(84,110)], [(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(57,64,71,78),(58,65,72,79),(59,66,73,80),(60,67,74,81),(61,68,75,82),(62,69,76,83),(63,70,77,84),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112)], [(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)], [(1,68,22,61,15,82,8,75),(2,81,23,74,16,67,9,60),(3,66,24,59,17,80,10,73),(4,79,25,72,18,65,11,58),(5,64,26,57,19,78,12,71),(6,77,27,70,20,63,13,84),(7,62,28,83,21,76,14,69),(29,101,50,94,43,87,36,108),(30,86,51,107,44,100,37,93),(31,99,52,92,45,85,38,106),(32,112,53,105,46,98,39,91),(33,97,54,90,47,111,40,104),(34,110,55,103,48,96,41,89),(35,95,56,88,49,109,42,102)]])
C2×C4.Dic7 is a maximal subgroup of
C28.8C42 C28.(C4⋊C4) C42⋊Dic7 C28.2C42 (C2×C28).Q8 C28.10C42 M4(2)⋊Dic7 (C2×C56)⋊C4 C28.4C42 M4(2)⋊4Dic7 C28.21C42 D14⋊2M4(2) Dic7⋊M4(2) C28⋊7M4(2) C4.Dic7⋊C4 C4○D28⋊C4 C28.(C2×Q8) C28.5C42 C42.43D14 C4⋊C4⋊36D14 (C2×C4).47D28 C42⋊4D14 C42.47D14 C28⋊3M4(2) C4⋊D4⋊D7 C7⋊C8⋊5D4 C7⋊C8⋊6D4 C7⋊C8.6D4 C28.12C42 Dic7⋊C8⋊C2 (C22×C8)⋊D7 M4(2)×Dic7 Dic7⋊4M4(2) C23.Dic14 M4(2).Dic7 D14⋊6M4(2) M4(2).31D14 C24.4Dic7 (D4×C14)⋊6C4 (Q8×C14)⋊6C4 C4○D4⋊Dic7 (D4×C14).11C4 (D4×C14)⋊9C4 (D4×C14).16C4 C2×D7×M4(2) C28.70C24 C28.76C24 C28.C24
C2×C4.Dic7 is a maximal quotient of
C28⋊7M4(2) C42.6Dic7 C42.7Dic7 C42.47D14 C28⋊3M4(2) C42.210D14 C24.4Dic7
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C4.Dic7 |
| kernel | C2×C4.Dic7 | C2×C7⋊C8 | C4.Dic7 | C22×C28 | C2×C28 | C22×C14 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 3 | 4 | 9 | 9 | 3 | 24 |
Matrix representation of C2×C4.Dic7 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 112 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 98 |
| 1 | 0 | 0 |
| 0 | 81 | 0 |
| 0 | 0 | 53 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 15 | 0 |
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[112,0,0,0,15,0,0,0,98],[1,0,0,0,81,0,0,0,53],[1,0,0,0,0,15,0,1,0] >;
C2×C4.Dic7 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_7 % in TeX
G:=Group("C2xC4.Dic7"); // GroupNames label
G:=SmallGroup(224,116);
// by ID
G=gap.SmallGroup(224,116);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,362,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations