direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D28⋊4C4, C23.50D28, M4(2)⋊23D14, C14⋊2C4≀C2, C4○D28⋊7C4, D28⋊21(C2×C4), (C2×D28)⋊15C4, (C2×C4).155D28, C28.445(C2×D4), (C2×C28).177D4, C4.14(D14⋊C4), (C2×Dic14)⋊15C4, Dic14⋊20(C2×C4), (C2×M4(2))⋊14D7, C28.69(C22×C4), C22.16(C2×D28), C28.68(C22⋊C4), (C14×M4(2))⋊22C2, (C2×C28).419C23, (C4×Dic7)⋊62C22, C4○D28.42C22, (C22×C14).106D4, (C22×C4).351D14, C22.52(D14⋊C4), (C7×M4(2))⋊35C22, (C22×C28).192C22, C7⋊3(C2×C4≀C2), C4.54(C2×C4×D7), (C2×C4×Dic7)⋊2C2, (C2×C4).85(C4×D7), C2.34(C2×D14⋊C4), (C2×C14).32(C2×D4), C4.136(C2×C7⋊D4), (C2×C28).112(C2×C4), (C2×C4○D28).14C2, C14.62(C2×C22⋊C4), (C2×C4).278(C7⋊D4), (C2×C4).512(C22×D7), (C2×C14).68(C22⋊C4), SmallGroup(448,672)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D28⋊4C4
G = < a,b,c,d | a2=b28=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b13, dcd-1=b19c >
Subgroups: 868 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C2×C4≀C2, C4×Dic7, C4×Dic7, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, D28⋊4C4, C2×C4×Dic7, C14×M4(2), C2×C4○D28, C2×D28⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4≀C2, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C4≀C2, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D28⋊4C4, C2×D14⋊C4, C2×D28⋊4C4
►On 112 points
Generators in S112
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 57)(25 58)(26 59)(27 60)(28 61)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(28 56)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 112)(81 111)(82 110)(83 109)(84 108)
(1 69 15 83)(2 82 16 68)(3 67 17 81)(4 80 18 66)(5 65 19 79)(6 78 20 64)(7 63 21 77)(8 76 22 62)(9 61 23 75)(10 74 24 60)(11 59 25 73)(12 72 26 58)(13 57 27 71)(14 70 28 84)(29 88)(30 101)(31 86)(32 99)(33 112)(34 97)(35 110)(36 95)(37 108)(38 93)(39 106)(40 91)(41 104)(42 89)(43 102)(44 87)(45 100)(46 85)(47 98)(48 111)(49 96)(50 109)(51 94)(52 107)(53 92)(54 105)(55 90)(56 103)
G:=sub<Sym(112)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,57)(25,58)(26,59)(27,60)(28,61)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,112)(81,111)(82,110)(83,109)(84,108), (1,69,15,83)(2,82,16,68)(3,67,17,81)(4,80,18,66)(5,65,19,79)(6,78,20,64)(7,63,21,77)(8,76,22,62)(9,61,23,75)(10,74,24,60)(11,59,25,73)(12,72,26,58)(13,57,27,71)(14,70,28,84)(29,88)(30,101)(31,86)(32,99)(33,112)(34,97)(35,110)(36,95)(37,108)(38,93)(39,106)(40,91)(41,104)(42,89)(43,102)(44,87)(45,100)(46,85)(47,98)(48,111)(49,96)(50,109)(51,94)(52,107)(53,92)(54,105)(55,90)(56,103)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,57)(25,58)(26,59)(27,60)(28,61)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,107)(58,106)(59,105)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,112)(81,111)(82,110)(83,109)(84,108), (1,69,15,83)(2,82,16,68)(3,67,17,81)(4,80,18,66)(5,65,19,79)(6,78,20,64)(7,63,21,77)(8,76,22,62)(9,61,23,75)(10,74,24,60)(11,59,25,73)(12,72,26,58)(13,57,27,71)(14,70,28,84)(29,88)(30,101)(31,86)(32,99)(33,112)(34,97)(35,110)(36,95)(37,108)(38,93)(39,106)(40,91)(41,104)(42,89)(43,102)(44,87)(45,100)(46,85)(47,98)(48,111)(49,96)(50,109)(51,94)(52,107)(53,92)(54,105)(55,90)(56,103) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,57),(25,58),(26,59),(27,60),(28,61),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(28,56),(57,107),(58,106),(59,105),(60,104),(61,103),(62,102),(63,101),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,112),(81,111),(82,110),(83,109),(84,108)], [(1,69,15,83),(2,82,16,68),(3,67,17,81),(4,80,18,66),(5,65,19,79),(6,78,20,64),(7,63,21,77),(8,76,22,62),(9,61,23,75),(10,74,24,60),(11,59,25,73),(12,72,26,58),(13,57,27,71),(14,70,28,84),(29,88),(30,101),(31,86),(32,99),(33,112),(34,97),(35,110),(36,95),(37,108),(38,93),(39,106),(40,91),(41,104),(42,89),(43,102),(44,87),(45,100),(46,85),(47,98),(48,111),(49,96),(50,109),(51,94),(52,107),(53,92),(54,105),(55,90),(56,103)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D7 | D14 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | D28 | D28⋊4C4 |
| kernel | C2×D28⋊4C4 | D28⋊4C4 | C2×C4×Dic7 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | M4(2) | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 3 | 6 | 3 | 8 | 12 | 6 | 12 | 6 | 12 |
Matrix representation of C2×D28⋊4C4 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 9 | 0 | 0 |
| 104 | 80 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 104 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 |
| 104 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,104,0,0,9,80,0,0,0,0,15,0,0,0,0,98],[1,0,0,0,104,112,0,0,0,0,0,1,0,0,1,0],[112,104,0,0,0,1,0,0,0,0,15,0,0,0,0,112] >;
C2×D28⋊4C4 in GAP, Magma, Sage, TeX
C_2\times D_{28}\rtimes_4C_4 % in TeX
G:=Group("C2xD28:4C4"); // GroupNames label
G:=SmallGroup(448,672);
// by ID
G=gap.SmallGroup(448,672);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,136,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^19*c>;
// generators/relations