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## G = D28.33C23order 448 = 26·7

### 14th non-split extension by D28 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D28.33C23
 Chief series C1 — C7 — C14 — C28 — D28 — C4○D28 — D4.10D14 — D28.33C23
 Lower central C7 — C14 — C28 — D28.33C23
 Upper central C1 — C2 — C4○D4 — 2+ 1+4

Generators and relations for D28.33C23
G = < a,b,c,d,e | a28=b2=1, c2=d2=e2=a14, bab=a-1, ac=ca, ad=da, eae-1=a15, bc=cb, bd=db, ebe-1=a7b, dcd-1=a14c, ce=ec, de=ed >

Subgroups: 980 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4○D4, Dic7, C28, C28, C28, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C7⋊C8, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4○SD16, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×Dic14, C4○D28, D42D7, Q8×D7, D4×C14, D4×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, D4.D14, C2×D4.D7, Q8.Dic7, D4.8D14, D4.9D14, D4.10D14, C7×2+ 1+4, D28.33C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C7⋊D4, C22×D7, D4○SD16, C2×C7⋊D4, C23×D7, C22×C7⋊D4, D28.33C23

Smallest permutation representation of D28.33C23
On 112 points
Generators in S112
```(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 84)(23 83)(24 82)(25 81)(26 80)(27 79)(28 78)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)
(1 39 15 53)(2 40 16 54)(3 41 17 55)(4 42 18 56)(5 43 19 29)(6 44 20 30)(7 45 21 31)(8 46 22 32)(9 47 23 33)(10 48 24 34)(11 49 25 35)(12 50 26 36)(13 51 27 37)(14 52 28 38)(57 96 71 110)(58 97 72 111)(59 98 73 112)(60 99 74 85)(61 100 75 86)(62 101 76 87)(63 102 77 88)(64 103 78 89)(65 104 79 90)(66 105 80 91)(67 106 81 92)(68 107 82 93)(69 108 83 94)(70 109 84 95)
(1 46 15 32)(2 47 16 33)(3 48 17 34)(4 49 18 35)(5 50 19 36)(6 51 20 37)(7 52 21 38)(8 53 22 39)(9 54 23 40)(10 55 24 41)(11 56 25 42)(12 29 26 43)(13 30 27 44)(14 31 28 45)(57 89 71 103)(58 90 72 104)(59 91 73 105)(60 92 74 106)(61 93 75 107)(62 94 76 108)(63 95 77 109)(64 96 78 110)(65 97 79 111)(66 98 80 112)(67 99 81 85)(68 100 82 86)(69 101 83 87)(70 102 84 88)
(1 53 15 39)(2 40 16 54)(3 55 17 41)(4 42 18 56)(5 29 19 43)(6 44 20 30)(7 31 21 45)(8 46 22 32)(9 33 23 47)(10 48 24 34)(11 35 25 49)(12 50 26 36)(13 37 27 51)(14 52 28 38)(57 103 71 89)(58 90 72 104)(59 105 73 91)(60 92 74 106)(61 107 75 93)(62 94 76 108)(63 109 77 95)(64 96 78 110)(65 111 79 97)(66 98 80 112)(67 85 81 99)(68 100 82 86)(69 87 83 101)(70 102 84 88)```

`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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99), (1,39,15,53)(2,40,16,54)(3,41,17,55)(4,42,18,56)(5,43,19,29)(6,44,20,30)(7,45,21,31)(8,46,22,32)(9,47,23,33)(10,48,24,34)(11,49,25,35)(12,50,26,36)(13,51,27,37)(14,52,28,38)(57,96,71,110)(58,97,72,111)(59,98,73,112)(60,99,74,85)(61,100,75,86)(62,101,76,87)(63,102,77,88)(64,103,78,89)(65,104,79,90)(66,105,80,91)(67,106,81,92)(68,107,82,93)(69,108,83,94)(70,109,84,95), (1,46,15,32)(2,47,16,33)(3,48,17,34)(4,49,18,35)(5,50,19,36)(6,51,20,37)(7,52,21,38)(8,53,22,39)(9,54,23,40)(10,55,24,41)(11,56,25,42)(12,29,26,43)(13,30,27,44)(14,31,28,45)(57,89,71,103)(58,90,72,104)(59,91,73,105)(60,92,74,106)(61,93,75,107)(62,94,76,108)(63,95,77,109)(64,96,78,110)(65,97,79,111)(66,98,80,112)(67,99,81,85)(68,100,82,86)(69,101,83,87)(70,102,84,88), (1,53,15,39)(2,40,16,54)(3,55,17,41)(4,42,18,56)(5,29,19,43)(6,44,20,30)(7,31,21,45)(8,46,22,32)(9,33,23,47)(10,48,24,34)(11,35,25,49)(12,50,26,36)(13,37,27,51)(14,52,28,38)(57,103,71,89)(58,90,72,104)(59,105,73,91)(60,92,74,106)(61,107,75,93)(62,94,76,108)(63,109,77,95)(64,96,78,110)(65,111,79,97)(66,98,80,112)(67,85,81,99)(68,100,82,86)(69,87,83,101)(70,102,84,88)>;`

`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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99), (1,39,15,53)(2,40,16,54)(3,41,17,55)(4,42,18,56)(5,43,19,29)(6,44,20,30)(7,45,21,31)(8,46,22,32)(9,47,23,33)(10,48,24,34)(11,49,25,35)(12,50,26,36)(13,51,27,37)(14,52,28,38)(57,96,71,110)(58,97,72,111)(59,98,73,112)(60,99,74,85)(61,100,75,86)(62,101,76,87)(63,102,77,88)(64,103,78,89)(65,104,79,90)(66,105,80,91)(67,106,81,92)(68,107,82,93)(69,108,83,94)(70,109,84,95), (1,46,15,32)(2,47,16,33)(3,48,17,34)(4,49,18,35)(5,50,19,36)(6,51,20,37)(7,52,21,38)(8,53,22,39)(9,54,23,40)(10,55,24,41)(11,56,25,42)(12,29,26,43)(13,30,27,44)(14,31,28,45)(57,89,71,103)(58,90,72,104)(59,91,73,105)(60,92,74,106)(61,93,75,107)(62,94,76,108)(63,95,77,109)(64,96,78,110)(65,97,79,111)(66,98,80,112)(67,99,81,85)(68,100,82,86)(69,101,83,87)(70,102,84,88), (1,53,15,39)(2,40,16,54)(3,55,17,41)(4,42,18,56)(5,29,19,43)(6,44,20,30)(7,31,21,45)(8,46,22,32)(9,33,23,47)(10,48,24,34)(11,35,25,49)(12,50,26,36)(13,37,27,51)(14,52,28,38)(57,103,71,89)(58,90,72,104)(59,105,73,91)(60,92,74,106)(61,107,75,93)(62,94,76,108)(63,109,77,95)(64,96,78,110)(65,111,79,97)(66,98,80,112)(67,85,81,99)(68,100,82,86)(69,87,83,101)(70,102,84,88) );`

`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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,84),(23,83),(24,82),(25,81),(26,80),(27,79),(28,78),(29,98),(30,97),(31,96),(32,95),(33,94),(34,93),(35,92),(36,91),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99)], [(1,39,15,53),(2,40,16,54),(3,41,17,55),(4,42,18,56),(5,43,19,29),(6,44,20,30),(7,45,21,31),(8,46,22,32),(9,47,23,33),(10,48,24,34),(11,49,25,35),(12,50,26,36),(13,51,27,37),(14,52,28,38),(57,96,71,110),(58,97,72,111),(59,98,73,112),(60,99,74,85),(61,100,75,86),(62,101,76,87),(63,102,77,88),(64,103,78,89),(65,104,79,90),(66,105,80,91),(67,106,81,92),(68,107,82,93),(69,108,83,94),(70,109,84,95)], [(1,46,15,32),(2,47,16,33),(3,48,17,34),(4,49,18,35),(5,50,19,36),(6,51,20,37),(7,52,21,38),(8,53,22,39),(9,54,23,40),(10,55,24,41),(11,56,25,42),(12,29,26,43),(13,30,27,44),(14,31,28,45),(57,89,71,103),(58,90,72,104),(59,91,73,105),(60,92,74,106),(61,93,75,107),(62,94,76,108),(63,95,77,109),(64,96,78,110),(65,97,79,111),(66,98,80,112),(67,99,81,85),(68,100,82,86),(69,101,83,87),(70,102,84,88)], [(1,53,15,39),(2,40,16,54),(3,55,17,41),(4,42,18,56),(5,29,19,43),(6,44,20,30),(7,31,21,45),(8,46,22,32),(9,33,23,47),(10,48,24,34),(11,35,25,49),(12,50,26,36),(13,37,27,51),(14,52,28,38),(57,103,71,89),(58,90,72,104),(59,105,73,91),(60,92,74,106),(61,107,75,93),(62,94,76,108),(63,109,77,95),(64,96,78,110),(65,111,79,97),(66,98,80,112),(67,85,81,99),(68,100,82,86),(69,87,83,101),(70,102,84,88)]])`

73 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 8E 14A 14B 14C 14D ··· 14AD 28A ··· 28R order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 14 14 14 14 ··· 14 28 ··· 28 size 1 1 2 2 2 4 4 4 28 2 2 2 2 4 28 28 28 2 2 2 14 14 28 28 28 2 2 2 4 ··· 4 4 ··· 4

73 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C7⋊D4 C7⋊D4 D4○SD16 D28.33C23 kernel D28.33C23 D4.D14 C2×D4.D7 Q8.Dic7 D4.8D14 D4.9D14 D4.10D14 C7×2+ 1+4 C7×D4 C7×Q8 2+ 1+4 C2×D4 C4○D4 D4 Q8 C7 C1 # reps 1 3 3 1 3 3 1 1 3 1 3 9 12 18 6 2 3

Matrix representation of D28.33C23 in GL6(𝔽113)

 28 0 0 0 0 0 8 109 0 0 0 0 0 0 57 90 0 0 0 0 102 56 0 0 0 0 28 89 112 77 0 0 76 71 44 1
,
 84 3 0 0 0 0 59 29 0 0 0 0 0 0 84 17 0 22 0 0 24 92 47 26 0 0 65 62 0 16 0 0 46 90 51 50
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 72 0 0 0 45 25 44 39 0 0 91 0 1 0 0 0 29 39 102 88
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 38 103 41 7 0 0 45 44 90 37 0 0 28 89 112 77 0 0 49 64 41 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 41 0 0 0 104 25 90 39 0 0 22 0 112 0 0 0 103 39 41 88

`G:=sub<GL(6,GF(113))| [28,8,0,0,0,0,0,109,0,0,0,0,0,0,57,102,28,76,0,0,90,56,89,71,0,0,0,0,112,44,0,0,0,0,77,1],[84,59,0,0,0,0,3,29,0,0,0,0,0,0,84,24,65,46,0,0,17,92,62,90,0,0,0,47,0,51,0,0,22,26,16,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,45,91,29,0,0,0,25,0,39,0,0,72,44,1,102,0,0,0,39,0,88],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,38,45,28,49,0,0,103,44,89,64,0,0,41,90,112,41,0,0,7,37,77,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,104,22,103,0,0,0,25,0,39,0,0,41,90,112,41,0,0,0,39,0,88] >;`

D28.33C23 in GAP, Magma, Sage, TeX

`D_{28}._{33}C_2^3`
`% in TeX`

`G:=Group("D28.33C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,1289);`
`// by ID`

`G=gap.SmallGroup(448,1289);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,136,1684,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^28=b^2=1,c^2=d^2=e^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^15,b*c=c*b,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^14*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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