direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D28, C28⋊5D4, C42⋊4D7, C7⋊1(C4×D4), C4⋊2(C4×D7), (C4×C28)⋊7C2, C28⋊4(C2×C4), D14⋊1(C2×C4), C14.2(C2×D4), C2.1(C2×D28), D14⋊C4⋊17C2, C4⋊Dic7⋊16C2, (C2×C4).75D14, (C2×D28).10C2, C14.4(C4○D4), C2.3(C4○D28), C14.4(C22×C4), (C2×C14).14C23, (C2×C28).86C22, C22.11(C22×D7), (C2×Dic7).25C22, (C22×D7).15C22, (C2×C4×D7)⋊7C2, C2.6(C2×C4×D7), SmallGroup(224,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D28
G = < a,b,c | a4=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 406 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C4×D4, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C4⋊Dic7, D14⋊C4, C4×C28, C2×C4×D7, C2×D28, C4×D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, D28, C22×D7, C2×C4×D7, C2×D28, C4○D28, C4×D28
►On 112 points
(1 66 51 95)(2 67 52 96)(3 68 53 97)(4 69 54 98)(5 70 55 99)(6 71 56 100)(7 72 29 101)(8 73 30 102)(9 74 31 103)(10 75 32 104)(11 76 33 105)(12 77 34 106)(13 78 35 107)(14 79 36 108)(15 80 37 109)(16 81 38 110)(17 82 39 111)(18 83 40 112)(19 84 41 85)(20 57 42 86)(21 58 43 87)(22 59 44 88)(23 60 45 89)(24 61 46 90)(25 62 47 91)(26 63 48 92)(27 64 49 93)(28 65 50 94)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(52 56)(53 55)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(82 84)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)
G:=sub<Sym(112)| (1,66,51,95)(2,67,52,96)(3,68,53,97)(4,69,54,98)(5,70,55,99)(6,71,56,100)(7,72,29,101)(8,73,30,102)(9,74,31,103)(10,75,32,104)(11,76,33,105)(12,77,34,106)(13,78,35,107)(14,79,36,108)(15,80,37,109)(16,81,38,110)(17,82,39,111)(18,83,40,112)(19,84,41,85)(20,57,42,86)(21,58,43,87)(22,59,44,88)(23,60,45,89)(24,61,46,90)(25,62,47,91)(26,63,48,92)(27,64,49,93)(28,65,50,94), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)>;
G:=Group( (1,66,51,95)(2,67,52,96)(3,68,53,97)(4,69,54,98)(5,70,55,99)(6,71,56,100)(7,72,29,101)(8,73,30,102)(9,74,31,103)(10,75,32,104)(11,76,33,105)(12,77,34,106)(13,78,35,107)(14,79,36,108)(15,80,37,109)(16,81,38,110)(17,82,39,111)(18,83,40,112)(19,84,41,85)(20,57,42,86)(21,58,43,87)(22,59,44,88)(23,60,45,89)(24,61,46,90)(25,62,47,91)(26,63,48,92)(27,64,49,93)(28,65,50,94), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99) );
G=PermutationGroup([[(1,66,51,95),(2,67,52,96),(3,68,53,97),(4,69,54,98),(5,70,55,99),(6,71,56,100),(7,72,29,101),(8,73,30,102),(9,74,31,103),(10,75,32,104),(11,76,33,105),(12,77,34,106),(13,78,35,107),(14,79,36,108),(15,80,37,109),(16,81,38,110),(17,82,39,111),(18,83,40,112),(19,84,41,85),(20,57,42,86),(21,58,43,87),(22,59,44,88),(23,60,45,89),(24,61,46,90),(25,62,47,91),(26,63,48,92),(27,64,49,93),(28,65,50,94)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(52,56),(53,55),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(82,84),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99)]])
C4×D28 is a maximal subgroup of
C4.17D56 D28⋊2C8 C8⋊6D28 C8⋊9D28 C42.16D14 D56⋊C4 D28⋊C8 D14⋊3M4(2) C28⋊2M4(2) C28⋊SD16 D28⋊3Q8 C4⋊D56 D28.19D4 D28⋊4Q8 D28.3Q8 C42.48D14 C42.56D14 D28.23D4 D28.4Q8 C28⋊2D8 C28⋊5SD16 D28⋊5Q8 D28⋊6Q8 C42.276D14 C42.277D14 C42⋊7D14 C42.91D14 C42⋊8D14 C42⋊10D14 C42.93D14 C42.95D14 C42.99D14 C42.100D14 C4×D4×D7 C42⋊11D14 C42⋊12D14 C42.228D14 D28⋊23D4 D28⋊24D4 D4⋊5D28 D4⋊6D28 C42.113D14 C42.116D14 C42.117D14 C42.119D14 C42.126D14 Q8⋊5D28 Q8⋊6D28 D28⋊10Q8 C42.131D14 C42.132D14 C42.133D14 C42.135D14 C42.136D14 D28⋊10D4 Dic14⋊10D4 C42⋊20D14 C42.143D14 D28⋊7Q8 C42.150D14 C42.152D14 C42.153D14 C42⋊23D14 C42⋊24D14 C42.161D14 C42.163D14 D28⋊11D4 Dic14⋊11D4 D28⋊12D4 D28⋊8Q8 D28⋊9Q8 C42.177D14 C42.179D14
C4×D28 is a maximal quotient of
C4⋊Dic7⋊7C4 (C2×C4)⋊9D28 D14⋊C4⋊C4 C2.(C4×D28) C8⋊6D28 D56⋊11C4 C8⋊9D28 C42.16D14 D56⋊C4 Dic28⋊C4 D56⋊4C4 C28⋊4(C4⋊C4) (C2×C4)⋊6D28 (C2×C42)⋊D7
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | C4×D7 | D28 | C4○D28 |
| kernel | C4×D28 | C4⋊Dic7 | D14⋊C4 | C4×C28 | C2×C4×D7 | C2×D28 | D28 | C28 | C42 | C14 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 9 | 12 | 12 | 12 |
Matrix representation of C4×D28 ►in GL3(𝔽29) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 28 | 0 | 0 |
| 0 | 2 | 24 |
| 0 | 5 | 17 |
| 28 | 0 | 0 |
| 0 | 21 | 8 |
| 0 | 3 | 8 |
G:=sub<GL(3,GF(29))| [12,0,0,0,12,0,0,0,12],[28,0,0,0,2,5,0,24,17],[28,0,0,0,21,3,0,8,8] >;
C4×D28 in GAP, Magma, Sage, TeX
C_4\times D_{28} % in TeX
G:=Group("C4xD28"); // GroupNames label
G:=SmallGroup(224,68);
// by ID
G=gap.SmallGroup(224,68);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^4=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations