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## G = C23.D14order 224 = 25·7

### 3rd non-split extension by C23 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C23.D14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — C23.D14
 Lower central C7 — C2×C14 — C23.D14
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C23.D14
G = < a,b,c,d,e | a2=b2=c2=1, d14=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d13 >

Subgroups: 198 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, Dic7, C28, C2×C14, C2×C14, C422C2, C2×Dic7, C2×C28, C22×C14, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C23.D14
Quotients: C1, C2, C22, C23, D7, C4○D4, D14, C422C2, C22×D7, C4○D28, D42D7, C23.D14

Smallest permutation representation of C23.D14
On 112 points
Generators in S112
```(2 64)(4 66)(6 68)(8 70)(10 72)(12 74)(14 76)(16 78)(18 80)(20 82)(22 84)(24 58)(26 60)(28 62)(29 100)(30 44)(31 102)(32 46)(33 104)(34 48)(35 106)(36 50)(37 108)(38 52)(39 110)(40 54)(41 112)(42 56)(43 86)(45 88)(47 90)(49 92)(51 94)(53 96)(55 98)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 85)
(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)
(1 109 77 38)(2 94 78 51)(3 107 79 36)(4 92 80 49)(5 105 81 34)(6 90 82 47)(7 103 83 32)(8 88 84 45)(9 101 57 30)(10 86 58 43)(11 99 59 56)(12 112 60 41)(13 97 61 54)(14 110 62 39)(15 95 63 52)(16 108 64 37)(17 93 65 50)(18 106 66 35)(19 91 67 48)(20 104 68 33)(21 89 69 46)(22 102 70 31)(23 87 71 44)(24 100 72 29)(25 85 73 42)(26 98 74 55)(27 111 75 40)(28 96 76 53)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 2 2 4 14 14 14 14 28 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 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 C4○D28 D4⋊2D7 kernel C23.D14 C4×Dic7 Dic7⋊C4 C4⋊Dic7 C23.D7 C7×C22⋊C4 C22⋊C4 C14 C2×C4 C23 C2 C2 # reps 1 1 2 1 2 1 3 6 6 3 12 6

Matrix representation of C23.D14 in GL4(𝔽29) generated by

 1 0 0 0 0 28 0 0 0 0 1 0 0 0 9 28
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 19 0 0 0 14 3
,
 17 0 0 0 0 17 0 0 0 0 23 11 0 0 23 6
`G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,9,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,19,14,0,0,0,3],[17,0,0,0,0,17,0,0,0,0,23,23,0,0,11,6] >;`

C23.D14 in GAP, Magma, Sage, TeX

`C_2^3.D_{14}`
`% in TeX`

`G:=Group("C2^3.D14");`
`// GroupNames label`

`G:=SmallGroup(224,74);`
`// by ID`

`G=gap.SmallGroup(224,74);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,506,188,6917]);`
`// Polycyclic`

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

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