direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4○D28, C14.4C24, D28⋊12C22, C28.43C23, D14.1C23, C23.26D14, Dic7.2C23, Dic14⋊11C22, (C2×C4)⋊10D14, (C22×C4)⋊6D7, (C2×D28)⋊14C2, C14⋊1(C4○D4), (C22×C28)⋊8C2, (C4×D7)⋊6C22, C7⋊D4⋊6C22, C2.5(C23×D7), (C2×C28)⋊13C22, C4.43(C22×D7), (C2×Dic14)⋊15C2, (C2×C14).65C23, C22.5(C22×D7), (C22×C14).46C22, (C2×Dic7).43C22, (C22×D7).28C22, C7⋊1(C2×C4○D4), (C2×C4×D7)⋊15C2, (C2×C7⋊D4)⋊12C2, SmallGroup(224,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4○D28
G = < a,b,c,d | a2=b4=d2=1, c14=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c13 >
Subgroups: 590 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C2×C4○D28
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C22×D7, C4○D28, C23×D7, C2×C4○D28
►On 112 points
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(1 97 15 111)(2 98 16 112)(3 99 17 85)(4 100 18 86)(5 101 19 87)(6 102 20 88)(7 103 21 89)(8 104 22 90)(9 105 23 91)(10 106 24 92)(11 107 25 93)(12 108 26 94)(13 109 27 95)(14 110 28 96)(29 71 43 57)(30 72 44 58)(31 73 45 59)(32 74 46 60)(33 75 47 61)(34 76 48 62)(35 77 49 63)(36 78 50 64)(37 79 51 65)(38 80 52 66)(39 81 53 67)(40 82 54 68)(41 83 55 69)(42 84 56 70)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(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)
G:=sub<Sym(112)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,97,15,111)(2,98,16,112)(3,99,17,85)(4,100,18,86)(5,101,19,87)(6,102,20,88)(7,103,21,89)(8,104,22,90)(9,105,23,91)(10,106,24,92)(11,107,25,93)(12,108,26,94)(13,109,27,95)(14,110,28,96)(29,71,43,57)(30,72,44,58)(31,73,45,59)(32,74,46,60)(33,75,47,61)(34,76,48,62)(35,77,49,63)(36,78,50,64)(37,79,51,65)(38,80,52,66)(39,81,53,67)(40,82,54,68)(41,83,55,69)(42,84,56,70), (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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(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)>;
G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,97,15,111)(2,98,16,112)(3,99,17,85)(4,100,18,86)(5,101,19,87)(6,102,20,88)(7,103,21,89)(8,104,22,90)(9,105,23,91)(10,106,24,92)(11,107,25,93)(12,108,26,94)(13,109,27,95)(14,110,28,96)(29,71,43,57)(30,72,44,58)(31,73,45,59)(32,74,46,60)(33,75,47,61)(34,76,48,62)(35,77,49,63)(36,78,50,64)(37,79,51,65)(38,80,52,66)(39,81,53,67)(40,82,54,68)(41,83,55,69)(42,84,56,70), (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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(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) );
G=PermutationGroup([[(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92)], [(1,97,15,111),(2,98,16,112),(3,99,17,85),(4,100,18,86),(5,101,19,87),(6,102,20,88),(7,103,21,89),(8,104,22,90),(9,105,23,91),(10,106,24,92),(11,107,25,93),(12,108,26,94),(13,109,27,95),(14,110,28,96),(29,71,43,57),(30,72,44,58),(31,73,45,59),(32,74,46,60),(33,75,47,61),(34,76,48,62),(35,77,49,63),(36,78,50,64),(37,79,51,65),(38,80,52,66),(39,81,53,67),(40,82,54,68),(41,83,55,69),(42,84,56,70)], [(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)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(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)]])
C2×C4○D28 is a maximal subgroup of
D14⋊C8⋊C2 D28.32D4 D28⋊14D4 C4○D28⋊C4 C4.(C2×D28) C42⋊4D14 (C2×D28)⋊13C4 D28⋊17D4 D28.37D4 (C22×C8)⋊D7 C23.23D28 (C2×D28).14C4 M4(2).31D14 C23.49D28 C23.20D28 C42.276D14 C24.27D14 C14.82+ 1+4 C14.2- 1+4 C14.2+ 1+4 C42.188D14 C42.91D14 C42⋊8D14 C42⋊9D14 C42.92D14 C42⋊12D14 C42.228D14 D28⋊23D4 D28⋊24D4 Dic14⋊23D4 Dic14⋊24D4 Dic14⋊20D4 C14.382+ 1+4 C14.722- 1+4 D28⋊20D4 C14.162- 1+4 C14.172- 1+4 D28⋊22D4 Dic14⋊22D4 C14.1212+ 1+4 C14.822- 1+4 C28.70C24 C56.9C23 C24.72D14 C24.41D14 C14.442- 1+4 C28.C24 (C2×C28)⋊17D4 C14.1082- 1+4 C2×D7×C4○D4 C14.C25
C2×C4○D28 is a maximal quotient of
C2×C4×Dic14 C42.274D14 C2×C4×D28 C42.276D14 C42.277D14 C24.27D14 C24.30D14 C24.31D14 C14.2- 1+4 C14.102+ 1+4 C14.52- 1+4 C14.112+ 1+4 C14.62- 1+4 C42.89D14 C42⋊10D14 C42.93D14 C42.94D14 C42.95D14 C42.96D14 C42.97D14 C42.98D14 C42.99D14 C42.100D14 C42.102D14 C42.104D14 C42.105D14 C42.106D14 C42⋊12D14 C42.228D14 D28⋊23D4 D28⋊24D4 Dic14⋊23D4 Dic14⋊24D4 C42⋊16D14 C42.229D14 C42.113D14 C42.114D14 C42⋊17D14 C42.115D14 C42.116D14 C42.117D14 C42.118D14 C42.119D14 Dic14⋊10Q8 C42.122D14 C42.232D14 D28⋊10Q8 C42.131D14 C42.132D14 C42.133D14 C42.134D14 C42.135D14 C42.136D14 C2×C4×C7⋊D4 C24.72D14
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | C4○D28 |
| kernel | C2×C4○D28 | C2×Dic14 | C2×C4×D7 | C2×D28 | C4○D28 | C2×C7⋊D4 | C22×C28 | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 3 | 4 | 18 | 3 | 24 |
Matrix representation of C2×C4○D28 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 25 | 5 |
| 0 | 12 | 21 |
| 1 | 0 | 0 |
| 0 | 27 | 16 |
| 0 | 27 | 2 |
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[1,0,0,0,12,0,0,0,12],[1,0,0,0,25,12,0,5,21],[1,0,0,0,27,27,0,16,2] >;
C2×C4○D28 in GAP, Magma, Sage, TeX
C_2\times C_4\circ D_{28} % in TeX
G:=Group("C2xC4oD28"); // GroupNames label
G:=SmallGroup(224,177);
// by ID
G=gap.SmallGroup(224,177);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,579,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^14=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^13>;
// generators/relations