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## G = C24.72D14order 448 = 26·7

### 12nd non-split extension by C24 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.72D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×C4○D28 — C24.72D14
 Lower central C7 — C2×C14 — C24.72D14
 Upper central C1 — C2×C4 — C23×C4

Generators and relations for C24.72D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 1284 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×C14, C22×C14, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, C22×C28, C23×C14, C28.48D4, C23.21D14, C4×C7⋊D4, C23.23D14, C287D4, C24⋊D7, C2×C4○D28, C23×C28, C24.72D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C7⋊D4, C22×D7, C22.19C24, C4○D28, C2×C7⋊D4, C23×D7, C2×C4○D28, C22×C7⋊D4, C24.72D14

Smallest permutation representation of C24.72D14
On 112 points
Generators in S112
```(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)
(57 104)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(65 112)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(57 104)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(65 112)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)
(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 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 75 15 61)(2 60 16 74)(3 73 17 59)(4 58 18 72)(5 71 19 57)(6 84 20 70)(7 69 21 83)(8 82 22 68)(9 67 23 81)(10 80 24 66)(11 65 25 79)(12 78 26 64)(13 63 27 77)(14 76 28 62)(29 95 43 109)(30 108 44 94)(31 93 45 107)(32 106 46 92)(33 91 47 105)(34 104 48 90)(35 89 49 103)(36 102 50 88)(37 87 51 101)(38 100 52 86)(39 85 53 99)(40 98 54 112)(41 111 55 97)(42 96 56 110)```

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 7A 7B 7C 14A ··· 14AS 28A ··· 28AV order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 28 28 1 1 1 1 2 ··· 2 28 ··· 28 2 2 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 C7⋊D4 C4○D28 kernel C24.72D14 C28.48D4 C23.21D14 C4×C7⋊D4 C23.23D14 C28⋊7D4 C24⋊D7 C2×C4○D28 C23×C28 C2×C28 C23×C4 C2×C14 C22×C4 C24 C2×C4 C22 # reps 1 2 1 4 2 2 2 1 1 4 3 8 18 3 24 48

Matrix representation of C24.72D14 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 28 0 0 0 0 1 0 0 0 0 28
,
 28 0 0 0 0 28 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
,
 26 0 0 0 0 10 0 0 0 0 9 0 0 0 0 13
,
 0 10 0 0 26 0 0 0 0 0 0 13 0 0 9 0
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[26,0,0,0,0,10,0,0,0,0,9,0,0,0,0,13],[0,26,0,0,10,0,0,0,0,0,0,9,0,0,13,0] >;`

C24.72D14 in GAP, Magma, Sage, TeX

`C_2^4._{72}D_{14}`
`% in TeX`

`G:=Group("C2^4.72D14");`
`// GroupNames label`

`G:=SmallGroup(448,1244);`
`// by ID`

`G=gap.SmallGroup(448,1244);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,18822]);`
`// Polycyclic`

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

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