metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).3D28, (C2×D28).3C4, (C2×C28).15D4, (C2×Q8).1D14, C4.10D4⋊5D7, (C4×Dic7).1C4, C28.10D4⋊1C2, C7⋊1(C42.C4), (Q8×C14).1C22, C14.13(C23⋊C4), C28.23D4.1C2, C22.14(D14⋊C4), C2.14(C23.1D14), (C2×C4).3(C4×D7), (C2×C28).3(C2×C4), (C2×C4).3(C7⋊D4), (C7×C4.10D4)⋊11C2, (C2×C14).7(C22⋊C4), SmallGroup(448,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).D28
G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, dad-1=ab14, cbc=b-1, dbd-1=ab15, dcd-1=b7c >
Subgroups: 428 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C4.10D4, C4.10D4, C4.4D4, C7⋊C8, C56, D28, C2×Dic7, C2×C28, C7×Q8, C22×D7, C42.C4, C4.Dic7, C4×Dic7, D14⋊C4, C7×M4(2), C2×D28, Q8×C14, C28.10D4, C7×C4.10D4, C28.23D4, (C2×C4).D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42.C4, D14⋊C4, C23.1D14, (C2×C4).D28
►On 112 points
Generators in S112
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)
(1 102 38 66 15 88 52 80)(2 89 53 67 16 103 39 81)(3 104 40 68 17 90 54 82)(4 91 55 69 18 105 41 83)(5 106 42 70 19 92 56 84)(6 93 29 71 20 107 43 57)(7 108 44 72 21 94 30 58)(8 95 31 73 22 109 45 59)(9 110 46 74 23 96 32 60)(10 97 33 75 24 111 47 61)(11 112 48 76 25 98 34 62)(12 99 35 77 26 85 49 63)(13 86 50 78 27 100 36 64)(14 101 37 79 28 87 51 65)
G:=sub<Sym(112)| (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104), (1,102,38,66,15,88,52,80)(2,89,53,67,16,103,39,81)(3,104,40,68,17,90,54,82)(4,91,55,69,18,105,41,83)(5,106,42,70,19,92,56,84)(6,93,29,71,20,107,43,57)(7,108,44,72,21,94,30,58)(8,95,31,73,22,109,45,59)(9,110,46,74,23,96,32,60)(10,97,33,75,24,111,47,61)(11,112,48,76,25,98,34,62)(12,99,35,77,26,85,49,63)(13,86,50,78,27,100,36,64)(14,101,37,79,28,87,51,65)>;
G:=Group( (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104), (1,102,38,66,15,88,52,80)(2,89,53,67,16,103,39,81)(3,104,40,68,17,90,54,82)(4,91,55,69,18,105,41,83)(5,106,42,70,19,92,56,84)(6,93,29,71,20,107,43,57)(7,108,44,72,21,94,30,58)(8,95,31,73,22,109,45,59)(9,110,46,74,23,96,32,60)(10,97,33,75,24,111,47,61)(11,112,48,76,25,98,34,62)(12,99,35,77,26,85,49,63)(13,86,50,78,27,100,36,64)(14,101,37,79,28,87,51,65) );
G=PermutationGroup([[(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(57,103),(58,102),(59,101),(60,100),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104)], [(1,102,38,66,15,88,52,80),(2,89,53,67,16,103,39,81),(3,104,40,68,17,90,54,82),(4,91,55,69,18,105,41,83),(5,106,42,70,19,92,56,84),(6,93,29,71,20,107,43,57),(7,108,44,72,21,94,30,58),(8,95,31,73,22,109,45,59),(9,110,46,74,23,96,32,60),(10,97,33,75,24,111,47,61),(11,112,48,76,25,98,34,62),(12,99,35,77,26,85,49,63),(13,86,50,78,27,100,36,64),(14,101,37,79,28,87,51,65)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 56 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C42.C4 | C23.1D14 | (C2×C4).D28 |
| kernel | (C2×C4).D28 | C28.10D4 | C7×C4.10D4 | C28.23D4 | C4×Dic7 | C2×D28 | C2×C28 | C4.10D4 | C2×Q8 | C2×C4 | C2×C4 | C2×C4 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 2 | 6 | 3 |
Matrix representation of (C2×C4).D28 ►in GL6(𝔽113)
| 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 | 112 | 0 |
| 0 | 0 | 22 | 0 | 0 | 112 |
| 112 | 24 | 0 | 0 | 0 | 0 |
| 89 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 43 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 104 | 0 | 34 | 98 |
| 112 | 24 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 107 | 62 | 0 | 0 |
| 0 | 0 | 45 | 6 | 0 | 0 |
| 0 | 0 | 60 | 0 | 37 | 87 |
| 0 | 0 | 2 | 4 | 70 | 76 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 102 | 0 | 102 | 1 |
| 0 | 0 | 90 | 87 | 0 | 0 |
| 0 | 0 | 89 | 76 | 11 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,22,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,89,0,0,0,0,24,10,0,0,0,0,0,0,15,43,0,104,0,0,0,98,0,0,0,0,0,0,15,34,0,0,0,0,0,98],[112,0,0,0,0,0,24,1,0,0,0,0,0,0,107,45,60,2,0,0,62,6,0,4,0,0,0,0,37,70,0,0,0,0,87,76],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,102,90,89,0,0,0,0,87,76,0,0,1,102,0,11,0,0,0,1,0,0] >;
(C2×C4).D28 in GAP, Magma, Sage, TeX
(C_2\times C_4).D_{28} % in TeX
G:=Group("(C2xC4).D28"); // GroupNames label
G:=SmallGroup(448,34);
// by ID
G=gap.SmallGroup(448,34);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,184,1123,794,297,136,851,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^2=1,d^4=b^14,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^14,c*b*c=b^-1,d*b*d^-1=a*b^15,d*c*d^-1=b^7*c>;
// generators/relations