direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28.46D4, M4(2)⋊22D14, (C2×C4).49D28, C4.65(C2×D28), (C2×D28).15C4, (C2×C28).172D4, C28.416(C2×D4), (C23×D7).3C4, C23.55(C4×D7), C4.28(D14⋊C4), C14⋊1(C4.D4), (C2×M4(2))⋊10D7, C28.53(C22⋊C4), (C14×M4(2))⋊18C2, (C2×C28).416C23, (C22×D28).14C2, (C22×C4).138D14, C4.Dic7⋊21C22, C22.50(D14⋊C4), (C2×D28).250C22, (C7×M4(2))⋊34C22, (C22×C28).187C22, C7⋊2(C2×C4.D4), (C2×C4).52(C4×D7), C22.20(C2×C4×D7), C2.29(C2×D14⋊C4), C4.109(C2×C7⋊D4), (C2×C28).107(C2×C4), C14.57(C2×C22⋊C4), (C2×C4.Dic7)⋊15C2, (C22×D7).5(C2×C4), (C2×C4).256(C7⋊D4), (C22×C14).70(C2×C4), (C2×C14).14(C22×C4), (C2×C4).120(C22×D7), (C2×C14).65(C22⋊C4), SmallGroup(448,664)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28.46D4
G = < a,b,c,d | a2=b28=d2=1, c4=b14, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b7c3 >
Subgroups: 1252 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C24, C28, D14, C2×C14, C2×C14, C4.D4, C2×M4(2), C2×M4(2), C22×D4, C7⋊C8, C56, D28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C4.D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C2×D28, C2×D28, C22×C28, C23×D7, C28.46D4, C2×C4.Dic7, C14×M4(2), C22×D28, C2×C28.46D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4.D4, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C4.D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C28.46D4, C2×D14⋊C4, C2×C28.46D4
►On 112 points
Generators in S112
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 29)(27 30)(28 31)(57 110)(58 111)(59 112)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)
(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)
(1 108 39 76 15 94 53 62)(2 107 40 75 16 93 54 61)(3 106 41 74 17 92 55 60)(4 105 42 73 18 91 56 59)(5 104 43 72 19 90 29 58)(6 103 44 71 20 89 30 57)(7 102 45 70 21 88 31 84)(8 101 46 69 22 87 32 83)(9 100 47 68 23 86 33 82)(10 99 48 67 24 85 34 81)(11 98 49 66 25 112 35 80)(12 97 50 65 26 111 36 79)(13 96 51 64 27 110 37 78)(14 95 52 63 28 109 38 77)
(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 67)(58 66)(59 65)(60 64)(61 63)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(85 89)(86 88)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)
G:=sub<Sym(112)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109), (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), (1,108,39,76,15,94,53,62)(2,107,40,75,16,93,54,61)(3,106,41,74,17,92,55,60)(4,105,42,73,18,91,56,59)(5,104,43,72,19,90,29,58)(6,103,44,71,20,89,30,57)(7,102,45,70,21,88,31,84)(8,101,46,69,22,87,32,83)(9,100,47,68,23,86,33,82)(10,99,48,67,24,85,34,81)(11,98,49,66,25,112,35,80)(12,97,50,65,26,111,36,79)(13,96,51,64,27,110,37,78)(14,95,52,63,28,109,38,77), (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)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109), (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), (1,108,39,76,15,94,53,62)(2,107,40,75,16,93,54,61)(3,106,41,74,17,92,55,60)(4,105,42,73,18,91,56,59)(5,104,43,72,19,90,29,58)(6,103,44,71,20,89,30,57)(7,102,45,70,21,88,31,84)(8,101,46,69,22,87,32,83)(9,100,47,68,23,86,33,82)(10,99,48,67,24,85,34,81)(11,98,49,66,25,112,35,80)(12,97,50,65,26,111,36,79)(13,96,51,64,27,110,37,78)(14,95,52,63,28,109,38,77), (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)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,29),(27,30),(28,31),(57,110),(58,111),(59,112),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109)], [(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)], [(1,108,39,76,15,94,53,62),(2,107,40,75,16,93,54,61),(3,106,41,74,17,92,55,60),(4,105,42,73,18,91,56,59),(5,104,43,72,19,90,29,58),(6,103,44,71,20,89,30,57),(7,102,45,70,21,88,31,84),(8,101,46,69,22,87,32,83),(9,100,47,68,23,86,33,82),(10,99,48,67,24,85,34,81),(11,98,49,66,25,112,35,80),(12,97,50,65,26,111,36,79),(13,96,51,64,27,110,37,78),(14,95,52,63,28,109,38,77)], [(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),(57,67),(58,66),(59,65),(60,64),(61,63),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(85,89),(86,88),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | C4.D4 | C28.46D4 |
| kernel | C2×C28.46D4 | C28.46D4 | C2×C4.Dic7 | C14×M4(2) | C22×D28 | C2×D28 | C23×D7 | C2×C28 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 6 | 3 | 6 | 12 | 12 | 6 | 2 | 12 |
Matrix representation of C2×C28.46D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 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 | 1 |
| 24 | 10 | 0 | 0 | 0 | 0 |
| 79 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 36 | 0 | 0 |
| 0 | 0 | 77 | 23 | 0 | 0 |
| 0 | 0 | 59 | 46 | 23 | 36 |
| 0 | 0 | 46 | 28 | 77 | 96 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 27 | 2 | 89 |
| 0 | 0 | 97 | 60 | 89 | 9 |
| 0 | 0 | 11 | 101 | 20 | 105 |
| 0 | 0 | 77 | 41 | 16 | 102 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 89 | 0 | 0 |
| 0 | 0 | 103 | 10 | 0 | 0 |
| 0 | 0 | 34 | 105 | 1 | 0 |
| 0 | 0 | 98 | 19 | 24 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,79,0,0,0,0,10,0,0,0,0,0,0,0,96,77,59,46,0,0,36,23,46,28,0,0,0,0,23,77,0,0,0,0,36,96],[0,34,0,0,0,0,10,0,0,0,0,0,0,0,44,97,11,77,0,0,27,60,101,41,0,0,2,89,20,16,0,0,89,9,105,102],[0,34,0,0,0,0,10,0,0,0,0,0,0,0,103,103,34,98,0,0,89,10,105,19,0,0,0,0,1,24,0,0,0,0,0,112] >;
C2×C28.46D4 in GAP, Magma, Sage, TeX
C_2\times C_{28}._{46}D_4 % in TeX
G:=Group("C2xC28.46D4"); // GroupNames label
G:=SmallGroup(448,664);
// by ID
G=gap.SmallGroup(448,664);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,1123,136,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=d^2=1,c^4=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^7*c^3>;
// generators/relations