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## G = C22.D28order 224 = 25·7

### 3rd non-split extension by C22 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C22.D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — C22.D28
 Lower central C7 — C2×C14 — C22.D28
 Upper central C1 — C22 — C22⋊C4

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, D42D7, C22.D28

Smallest permutation representation of C22.D28
On 112 points
Generators in S112
```(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)]])`

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`

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