metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D28, C23.16D14, D14⋊C4⋊7C2, C4⋊Dic7⋊5C2, C22⋊C4⋊6D7, C2.8(C2×D28), (C2×C14).4D4, (C2×C4).9D14, C14.6(C2×D4), (C2×C28).3C22, C14.23(C4○D4), (C2×C14).27C23, (C22×Dic7)⋊2C2, C7⋊2(C22.D4), C2.10(D4⋊2D7), (C22×D7).5C22, C22.45(C22×D7), (C22×C14).16C22, (C2×Dic7).28C22, (C7×C22⋊C4)⋊4C2, (C2×C7⋊D4).5C2, SmallGroup(224,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D28
G = < a,b,c,d | a2=b2=c28=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 326 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C22.D4, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C22×Dic7, C2×C7⋊D4, C22.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, D28, C22×D7, C2×D28, D4⋊2D7, C22.D28
►On 112 points
(1 111)(2 71)(3 85)(4 73)(5 87)(6 75)(7 89)(8 77)(9 91)(10 79)(11 93)(12 81)(13 95)(14 83)(15 97)(16 57)(17 99)(18 59)(19 101)(20 61)(21 103)(22 63)(23 105)(24 65)(25 107)(26 67)(27 109)(28 69)(29 72)(30 86)(31 74)(32 88)(33 76)(34 90)(35 78)(36 92)(37 80)(38 94)(39 82)(40 96)(41 84)(42 98)(43 58)(44 100)(45 60)(46 102)(47 62)(48 104)(49 64)(50 106)(51 66)(52 108)(53 68)(54 110)(55 70)(56 112)
(1 55)(2 56)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)
(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 54 55 28)(2 27 56 53)(3 52 29 26)(4 25 30 51)(5 50 31 24)(6 23 32 49)(7 48 33 22)(8 21 34 47)(9 46 35 20)(10 19 36 45)(11 44 37 18)(12 17 38 43)(13 42 39 16)(14 15 40 41)(57 95 98 82)(58 81 99 94)(59 93 100 80)(60 79 101 92)(61 91 102 78)(62 77 103 90)(63 89 104 76)(64 75 105 88)(65 87 106 74)(66 73 107 86)(67 85 108 72)(68 71 109 112)(69 111 110 70)(83 97 96 84)
G:=sub<Sym(112)| (1,111)(2,71)(3,85)(4,73)(5,87)(6,75)(7,89)(8,77)(9,91)(10,79)(11,93)(12,81)(13,95)(14,83)(15,97)(16,57)(17,99)(18,59)(19,101)(20,61)(21,103)(22,63)(23,105)(24,65)(25,107)(26,67)(27,109)(28,69)(29,72)(30,86)(31,74)(32,88)(33,76)(34,90)(35,78)(36,92)(37,80)(38,94)(39,82)(40,96)(41,84)(42,98)(43,58)(44,100)(45,60)(46,102)(47,62)(48,104)(49,64)(50,106)(51,66)(52,108)(53,68)(54,110)(55,70)(56,112), (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97), (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), (1,54,55,28)(2,27,56,53)(3,52,29,26)(4,25,30,51)(5,50,31,24)(6,23,32,49)(7,48,33,22)(8,21,34,47)(9,46,35,20)(10,19,36,45)(11,44,37,18)(12,17,38,43)(13,42,39,16)(14,15,40,41)(57,95,98,82)(58,81,99,94)(59,93,100,80)(60,79,101,92)(61,91,102,78)(62,77,103,90)(63,89,104,76)(64,75,105,88)(65,87,106,74)(66,73,107,86)(67,85,108,72)(68,71,109,112)(69,111,110,70)(83,97,96,84)>;
G:=Group( (1,111)(2,71)(3,85)(4,73)(5,87)(6,75)(7,89)(8,77)(9,91)(10,79)(11,93)(12,81)(13,95)(14,83)(15,97)(16,57)(17,99)(18,59)(19,101)(20,61)(21,103)(22,63)(23,105)(24,65)(25,107)(26,67)(27,109)(28,69)(29,72)(30,86)(31,74)(32,88)(33,76)(34,90)(35,78)(36,92)(37,80)(38,94)(39,82)(40,96)(41,84)(42,98)(43,58)(44,100)(45,60)(46,102)(47,62)(48,104)(49,64)(50,106)(51,66)(52,108)(53,68)(54,110)(55,70)(56,112), (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97), (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), (1,54,55,28)(2,27,56,53)(3,52,29,26)(4,25,30,51)(5,50,31,24)(6,23,32,49)(7,48,33,22)(8,21,34,47)(9,46,35,20)(10,19,36,45)(11,44,37,18)(12,17,38,43)(13,42,39,16)(14,15,40,41)(57,95,98,82)(58,81,99,94)(59,93,100,80)(60,79,101,92)(61,91,102,78)(62,77,103,90)(63,89,104,76)(64,75,105,88)(65,87,106,74)(66,73,107,86)(67,85,108,72)(68,71,109,112)(69,111,110,70)(83,97,96,84) );
G=PermutationGroup([[(1,111),(2,71),(3,85),(4,73),(5,87),(6,75),(7,89),(8,77),(9,91),(10,79),(11,93),(12,81),(13,95),(14,83),(15,97),(16,57),(17,99),(18,59),(19,101),(20,61),(21,103),(22,63),(23,105),(24,65),(25,107),(26,67),(27,109),(28,69),(29,72),(30,86),(31,74),(32,88),(33,76),(34,90),(35,78),(36,92),(37,80),(38,94),(39,82),(40,96),(41,84),(42,98),(43,58),(44,100),(45,60),(46,102),(47,62),(48,104),(49,64),(50,106),(51,66),(52,108),(53,68),(54,110),(55,70),(56,112)], [(1,55),(2,56),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97)], [(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)], [(1,54,55,28),(2,27,56,53),(3,52,29,26),(4,25,30,51),(5,50,31,24),(6,23,32,49),(7,48,33,22),(8,21,34,47),(9,46,35,20),(10,19,36,45),(11,44,37,18),(12,17,38,43),(13,42,39,16),(14,15,40,41),(57,95,98,82),(58,81,99,94),(59,93,100,80),(60,79,101,92),(61,91,102,78),(62,77,103,90),(63,89,104,76),(64,75,105,88),(65,87,106,74),(66,73,107,86),(67,85,108,72),(68,71,109,112),(69,111,110,70),(83,97,96,84)]])
C22.D28 is a maximal subgroup of
C23.5D28 C23⋊3D28 C24.31D14 C42⋊8D14 C42.92D14 C42.96D14 C42.102D14 D4⋊5D28 D4⋊6D28 C42.118D14 C24.56D14 C24⋊2D14 C24.33D14 C14.462+ 1+4 C14.1152+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C22⋊Q8⋊25D7 C14.532+ 1+4 C14.772- 1+4 C14.572+ 1+4 C14.792- 1+4 D7×C22.D4 C14.822- 1+4 C14.1222+ 1+4 C14.662+ 1+4 C14.852- 1+4 C14.862- 1+4 C42⋊20D14 C42.143D14 C42.144D14 C42.145D14 C42.161D14 C42.163D14 C42.164D14 C42.165D14
C22.D28 is a maximal quotient of
C4⋊Dic7⋊7C4 (C2×C28).28D4 C14.(C4⋊Q8) C2.(C4×D28) (C2×C4).21D28 (C2×C28).33D4 C23.34D28 C23.35D28 C23.10D28 C23.38D28 C22.D56 C23.13D28 C23.42D28 C24.47D14 C24.10D14 C23.45D28 C23.16D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D28 | D4⋊2D7 |
| kernel | C22.D28 | C4⋊Dic7 | D14⋊C4 | C7×C22⋊C4 | C22×Dic7 | C2×C7⋊D4 | C2×C14 | C22⋊C4 | C14 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 6 |
Matrix representation of C22.D28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 27 |
| 0 | 0 | 28 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 20 | 25 | 0 | 0 |
| 4 | 21 | 0 | 0 |
| 0 | 0 | 17 | 27 |
| 0 | 0 | 0 | 12 |
| 9 | 4 | 0 | 0 |
| 9 | 20 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,17,28,0,0,27,12],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[20,4,0,0,25,21,0,0,0,0,17,0,0,0,27,12],[9,9,0,0,4,20,0,0,0,0,17,0,0,0,0,17] >;
C22.D28 in GAP, Magma, Sage, TeX
C_2^2.D_{28} % in TeX
G:=Group("C2^2.D28"); // GroupNames label
G:=SmallGroup(224,81);
// by ID
G=gap.SmallGroup(224,81);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,188,122,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations