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## G = D28.34C23order 448 = 26·7

### 15th non-split extension by D28 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D28.34C23
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — D4⋊8D14 — D28.34C23
 Lower central C7 — C14 — C28 — D28.34C23
 Upper central C1 — C2 — C4○D4 — 2- 1+4

Generators and relations for D28.34C23
G = < a,b,c,d,e | a28=b2=c2=d2=1, e2=a14, bab=dad=a-1, ac=ca, eae-1=a15, cbc=a14b, dbd=a26b, ebe-1=a7b, cd=dc, ce=ec, ede-1=a21d >

Subgroups: 1172 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, 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, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, D4○SD16, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×D28, C4○D28, D4×D7, Q82D7, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C2×Q8⋊D7, C28.C23, Q8.Dic7, D4⋊D14, D4.8D14, D48D14, C7×2- 1+4, D28.34C23
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, D28.34C23

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

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

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

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

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 ··· 14R 28A ··· 28AD 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 28 28 28 2 2 2 2 4 4 4 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 D28.34C23 kernel D28.34C23 C2×Q8⋊D7 C28.C23 Q8.Dic7 D4⋊D14 D4.8D14 D4⋊8D14 C7×2- 1+4 C7×D4 C7×Q8 2- 1+4 C2×Q8 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 D28.34C23 in GL6(𝔽113)

 25 34 0 0 0 0 79 112 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 112 112 1 2 0 0 0 1 112 112
,
 25 34 0 0 0 0 88 88 0 0 0 0 0 0 0 112 0 0 0 0 112 0 0 0 0 0 0 0 112 0 0 0 112 112 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 100 100 26 26 0 0 100 100 0 26 0 0 100 13 0 0 0 0 100 87 13 26
,
 112 0 0 0 0 0 34 1 0 0 0 0 0 0 100 100 0 26 0 0 100 100 26 26 0 0 13 100 0 0 0 0 87 100 13 26
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 1 1 112 111 0 0 112 0 0 0 0 0 1 1 0 112

`G:=sub<GL(6,GF(113))| [25,79,0,0,0,0,34,112,0,0,0,0,0,0,0,112,112,0,0,0,1,0,112,1,0,0,0,0,1,112,0,0,0,0,2,112],[25,88,0,0,0,0,34,88,0,0,0,0,0,0,0,112,0,112,0,0,112,0,0,112,0,0,0,0,112,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,100,100,100,0,0,100,100,13,87,0,0,26,0,0,13,0,0,26,26,0,26],[112,34,0,0,0,0,0,1,0,0,0,0,0,0,100,100,13,87,0,0,100,100,100,100,0,0,0,26,0,13,0,0,26,26,0,26],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,112,1,0,0,0,1,0,1,0,0,1,112,0,0,0,0,0,111,0,112] >;`

D28.34C23 in GAP, Magma, Sage, TeX

`D_{28}._{34}C_2^3`
`% in TeX`

`G:=Group("D28.34C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,1290);`
`// by ID`

`G=gap.SmallGroup(448,1290);`
`# by ID`

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

`G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=1,e^2=a^14,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^15,c*b*c=a^14*b,d*b*d=a^26*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^21*d>;`
`// generators/relations`

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