direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C28, C42⋊4C14, C4⋊C4⋊7C14, C4⋊1(C2×C28), C28⋊6(C2×C4), (C4×C28)⋊11C2, C2.3(D4×C14), C22⋊C4⋊6C14, (C22×C28)⋊4C2, C22⋊1(C2×C28), (C22×C4)⋊2C14, (C2×D4).7C14, C14.66(C2×D4), (D4×C14).14C2, C2.4(C22×C28), C23.7(C2×C14), C14.39(C4○D4), (C2×C14).73C23, C14.32(C22×C4), (C2×C28).121C22, C22.7(C22×C14), (C22×C14).26C22, (C7×C4⋊C4)⋊16C2, (C2×C14)⋊4(C2×C4), (C2×C28)○(D4×C14), C2.2(C7×C4○D4), (C7×C22⋊C4)⋊14C2, (C2×C4).15(C2×C14), (C2×C28)○(C7×C4⋊C4), (C2×C28)○(C7×C22⋊C4), SmallGroup(224,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C28
G = < a,b,c | a28=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C2×C14, C4×D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×C28, C7×D4, C22×C14, C22×C28, D4×C14, C7×C4○D4, D4×C28
►On 112 points
(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 67 30)(2 109 68 31)(3 110 69 32)(4 111 70 33)(5 112 71 34)(6 85 72 35)(7 86 73 36)(8 87 74 37)(9 88 75 38)(10 89 76 39)(11 90 77 40)(12 91 78 41)(13 92 79 42)(14 93 80 43)(15 94 81 44)(16 95 82 45)(17 96 83 46)(18 97 84 47)(19 98 57 48)(20 99 58 49)(21 100 59 50)(22 101 60 51)(23 102 61 52)(24 103 62 53)(25 104 63 54)(26 105 64 55)(27 106 65 56)(28 107 66 29)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(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)
G:=sub<Sym(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), (1,108,67,30)(2,109,68,31)(3,110,69,32)(4,111,70,33)(5,112,71,34)(6,85,72,35)(7,86,73,36)(8,87,74,37)(9,88,75,38)(10,89,76,39)(11,90,77,40)(12,91,78,41)(13,92,79,42)(14,93,80,43)(15,94,81,44)(16,95,82,45)(17,96,83,46)(18,97,84,47)(19,98,57,48)(20,99,58,49)(21,100,59,50)(22,101,60,51)(23,102,61,52)(24,103,62,53)(25,104,63,54)(26,105,64,55)(27,106,65,56)(28,107,66,29), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(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)>;
G:=Group( (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), (1,108,67,30)(2,109,68,31)(3,110,69,32)(4,111,70,33)(5,112,71,34)(6,85,72,35)(7,86,73,36)(8,87,74,37)(9,88,75,38)(10,89,76,39)(11,90,77,40)(12,91,78,41)(13,92,79,42)(14,93,80,43)(15,94,81,44)(16,95,82,45)(17,96,83,46)(18,97,84,47)(19,98,57,48)(20,99,58,49)(21,100,59,50)(22,101,60,51)(23,102,61,52)(24,103,62,53)(25,104,63,54)(26,105,64,55)(27,106,65,56)(28,107,66,29), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(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) );
G=PermutationGroup([[(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)], [(1,108,67,30),(2,109,68,31),(3,110,69,32),(4,111,70,33),(5,112,71,34),(6,85,72,35),(7,86,73,36),(8,87,74,37),(9,88,75,38),(10,89,76,39),(11,90,77,40),(12,91,78,41),(13,92,79,42),(14,93,80,43),(15,94,81,44),(16,95,82,45),(17,96,83,46),(18,97,84,47),(19,98,57,48),(20,99,58,49),(21,100,59,50),(22,101,60,51),(23,102,61,52),(24,103,62,53),(25,104,63,54),(26,105,64,55),(27,106,65,56),(28,107,66,29)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(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)]])
D4×C28 is a maximal subgroup of
C28.57D8 C28.50D8 C28.38SD16 D4.3Dic14 C42.47D14 C28⋊3M4(2) C42.48D14 C28⋊7D8 D4.1D28 C42.51D14 D4.2D28 C42.102D14 D4⋊5Dic14 C42.104D14 C42.105D14 C42.106D14 D4⋊6Dic14 C42⋊11D14 C42.108D14 C42⋊12D14 C42.228D14 D28⋊23D4 D28⋊24D4 Dic14⋊23D4 Dic14⋊24D4 D4⋊5D28 D4⋊6D28 C42⋊16D14 C42.229D14 C42.113D14 C42.114D14 C42⋊17D14 C42.115D14 C42.116D14 C42.117D14 C42.118D14 C42.119D14
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | D4×C28 | C4×C28 | C7×C22⋊C4 | C7×C4⋊C4 | C22×C28 | D4×C14 | C7×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C28 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 2 | 12 | 12 |
Matrix representation of D4×C28 ►in GL3(𝔽29) generated by
| 12 | 0 | 0 |
| 0 | 20 | 0 |
| 0 | 0 | 20 |
| 1 | 0 | 0 |
| 0 | 1 | 2 |
| 0 | 28 | 28 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 28 | 28 |
G:=sub<GL(3,GF(29))| [12,0,0,0,20,0,0,0,20],[1,0,0,0,1,28,0,2,28],[1,0,0,0,1,28,0,0,28] >;
D4×C28 in GAP, Magma, Sage, TeX
D_4\times C_{28} % in TeX
G:=Group("D4xC28"); // GroupNames label
G:=SmallGroup(224,153);
// by ID
G=gap.SmallGroup(224,153);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,518]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations