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## G = D4.10D14order 224 = 25·7

### The non-split extension by D4 of D14 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4.10D14
 Chief series C1 — C7 — C14 — D14 — C4×D7 — Q8×D7 — D4.10D14
 Lower central C7 — C14 — D4.10D14
 Upper central C1 — C2 — C4○D4

Generators and relations for D4.10D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c13 >

Subgroups: 486 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C2×Dic14, C4○D28, D42D7, Q8×D7, C7×C4○D4, D4.10D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C22×D7, C23×D7, D4.10D14

Smallest permutation representation of D4.10D14
On 112 points
Generators in S112
```(1 89 15 103)(2 104 16 90)(3 91 17 105)(4 106 18 92)(5 93 19 107)(6 108 20 94)(7 95 21 109)(8 110 22 96)(9 97 23 111)(10 112 24 98)(11 99 25 85)(12 86 26 100)(13 101 27 87)(14 88 28 102)(29 81 43 67)(30 68 44 82)(31 83 45 69)(32 70 46 84)(33 57 47 71)(34 72 48 58)(35 59 49 73)(36 74 50 60)(37 61 51 75)(38 76 52 62)(39 63 53 77)(40 78 54 64)(41 65 55 79)(42 80 56 66)
(1 54)(2 41)(3 56)(4 43)(5 30)(6 45)(7 32)(8 47)(9 34)(10 49)(11 36)(12 51)(13 38)(14 53)(15 40)(16 55)(17 42)(18 29)(19 44)(20 31)(21 46)(22 33)(23 48)(24 35)(25 50)(26 37)(27 52)(28 39)(57 110)(58 97)(59 112)(60 99)(61 86)(62 101)(63 88)(64 103)(65 90)(66 105)(67 92)(68 107)(69 94)(70 109)(71 96)(72 111)(73 98)(74 85)(75 100)(76 87)(77 102)(78 89)(79 104)(80 91)(81 106)(82 93)(83 108)(84 95)
(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 70 15 84)(2 83 16 69)(3 68 17 82)(4 81 18 67)(5 66 19 80)(6 79 20 65)(7 64 21 78)(8 77 22 63)(9 62 23 76)(10 75 24 61)(11 60 25 74)(12 73 26 59)(13 58 27 72)(14 71 28 57)(29 92 43 106)(30 105 44 91)(31 90 45 104)(32 103 46 89)(33 88 47 102)(34 101 48 87)(35 86 49 100)(36 99 50 85)(37 112 51 98)(38 97 52 111)(39 110 53 96)(40 95 54 109)(41 108 55 94)(42 93 56 107)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4J 7A 7B 7C 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 7 7 7 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 14 14 2 2 2 2 14 ··· 14 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D7 D14 D14 D14 2- 1+4 D4.10D14 kernel D4.10D14 C2×Dic14 C4○D28 D4⋊2D7 Q8×D7 C7×C4○D4 C4○D4 C2×C4 D4 Q8 C7 C1 # reps 1 3 3 6 2 1 3 9 9 3 1 6

Matrix representation of D4.10D14 in GL4(𝔽29) generated by

 6 0 28 28 0 6 2 3 24 8 23 0 13 21 0 23
,
 4 19 28 1 21 25 0 24 8 19 15 10 0 19 19 14
,
 11 8 19 3 18 0 7 7 7 5 26 21 11 16 8 21
,
 16 18 18 11 10 13 26 0 0 20 18 27 13 20 2 11
`G:=sub<GL(4,GF(29))| [6,0,24,13,0,6,8,21,28,2,23,0,28,3,0,23],[4,21,8,0,19,25,19,19,28,0,15,19,1,24,10,14],[11,18,7,11,8,0,5,16,19,7,26,8,3,7,21,21],[16,10,0,13,18,13,20,20,18,26,18,2,11,0,27,11] >;`

D4.10D14 in GAP, Magma, Sage, TeX

`D_4._{10}D_{14}`
`% in TeX`

`G:=Group("D4.10D14");`
`// GroupNames label`

`G:=SmallGroup(224,186);`
`// by ID`

`G=gap.SmallGroup(224,186);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,86,579,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^14=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^13>;`
`// generators/relations`

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