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(C22.2D28), (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×D28).C4
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 [×2], C4 [×4], C22, C22 [×3], C7, C8 [×2], C2×C4 [×3], C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, Dic7, C28 [×3], D14 [×3], C2×C14, C4.10D4, C4.10D4, C4.4D4, C7⋊C8, C56, D28, C2×Dic7, C2×C28 [×3], C7×Q8, C22×D7, C42.C4, C4.Dic7, C4×Dic7, D14⋊C4 [×2], C7×M4(2), C2×D28, Q8×C14, C28.10D4, C7×C4.10D4, C28.23D4, (C2×D28).C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42.C4, D14⋊C4, C22.2D28, (C2×D28).C4
►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 32)(2 31)(3 30)(4 29)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)
(1 86 33 79 15 100 47 65)(2 101 48 80 16 87 34 66)(3 88 35 81 17 102 49 67)(4 103 50 82 18 89 36 68)(5 90 37 83 19 104 51 69)(6 105 52 84 20 91 38 70)(7 92 39 57 21 106 53 71)(8 107 54 58 22 93 40 72)(9 94 41 59 23 108 55 73)(10 109 56 60 24 95 42 74)(11 96 43 61 25 110 29 75)(12 111 30 62 26 97 44 76)(13 98 45 63 27 112 31 77)(14 85 32 64 28 99 46 78)
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), (1,32)(2,31)(3,30)(4,29)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101), (1,86,33,79,15,100,47,65)(2,101,48,80,16,87,34,66)(3,88,35,81,17,102,49,67)(4,103,50,82,18,89,36,68)(5,90,37,83,19,104,51,69)(6,105,52,84,20,91,38,70)(7,92,39,57,21,106,53,71)(8,107,54,58,22,93,40,72)(9,94,41,59,23,108,55,73)(10,109,56,60,24,95,42,74)(11,96,43,61,25,110,29,75)(12,111,30,62,26,97,44,76)(13,98,45,63,27,112,31,77)(14,85,32,64,28,99,46,78)>;
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), (1,32)(2,31)(3,30)(4,29)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101), (1,86,33,79,15,100,47,65)(2,101,48,80,16,87,34,66)(3,88,35,81,17,102,49,67)(4,103,50,82,18,89,36,68)(5,90,37,83,19,104,51,69)(6,105,52,84,20,91,38,70)(7,92,39,57,21,106,53,71)(8,107,54,58,22,93,40,72)(9,94,41,59,23,108,55,73)(10,109,56,60,24,95,42,74)(11,96,43,61,25,110,29,75)(12,111,30,62,26,97,44,76)(13,98,45,63,27,112,31,77)(14,85,32,64,28,99,46,78) );
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)], [(1,32),(2,31),(3,30),(4,29),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(57,100),(58,99),(59,98),(60,97),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101)], [(1,86,33,79,15,100,47,65),(2,101,48,80,16,87,34,66),(3,88,35,81,17,102,49,67),(4,103,50,82,18,89,36,68),(5,90,37,83,19,104,51,69),(6,105,52,84,20,91,38,70),(7,92,39,57,21,106,53,71),(8,107,54,58,22,93,40,72),(9,94,41,59,23,108,55,73),(10,109,56,60,24,95,42,74),(11,96,43,61,25,110,29,75),(12,111,30,62,26,97,44,76),(13,98,45,63,27,112,31,77),(14,85,32,64,28,99,46,78)])
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 | C22.2D28 | (C2×D28).C4 |
| kernel | (C2×D28).C4 | 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×D28).C4 ►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×D28).C4 in GAP, Magma, Sage, TeX
(C_2\times D_{28}).C_4 % in TeX
G:=Group("(C2xD28).C4"); // 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