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## G = C28.76C24order 448 = 26·7

### 23rd non-split extension by C28 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C28.76C24
 Chief series C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — Q8.Dic7 — C28.76C24
 Lower central C7 — C14 — C28.76C24
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C28.76C24
G = < a,b,c,d,e | a28=c2=d2=e2=1, b2=a21, bab-1=a13, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a14b, dcd=a14c, ce=ec, de=ed >

Subgroups: 596 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C8○D4, C2×C4○D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, Q8○M4(2), C2×C7⋊C8, C4.Dic7, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C2×C4.Dic7, Q8.Dic7, C14×C4○D4, C28.76C24
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, Dic7, D14, C23×C4, C2×Dic7, C22×D7, Q8○M4(2), C22×Dic7, C23×D7, C23×Dic7, C28.76C24

Smallest permutation representation of C28.76C24
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 84 22 77 15 70 8 63)(2 69 23 62 16 83 9 76)(3 82 24 75 17 68 10 61)(4 67 25 60 18 81 11 74)(5 80 26 73 19 66 12 59)(6 65 27 58 20 79 13 72)(7 78 28 71 21 64 14 57)(29 96 50 89 43 110 36 103)(30 109 51 102 44 95 37 88)(31 94 52 87 45 108 38 101)(32 107 53 100 46 93 39 86)(33 92 54 85 47 106 40 99)(34 105 55 98 48 91 41 112)(35 90 56 111 49 104 42 97)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)
(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)(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)
(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)```

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

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

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

94 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 7A 7B 7C 8A ··· 8P 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 ··· 2 4 4 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 ··· 2 1 1 2 ··· 2 2 2 2 14 ··· 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

94 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - - - + image C1 C2 C2 C2 C4 C4 C4 D7 D14 Dic7 Dic7 Dic7 D14 Q8○M4(2) C28.76C24 kernel C28.76C24 C2×C4.Dic7 Q8.Dic7 C14×C4○D4 D4×C14 Q8×C14 C7×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C7 C1 # reps 1 6 8 1 6 2 8 3 9 9 3 12 12 2 12

Matrix representation of C28.76C24 in GL4(𝔽113) generated by

 99 0 0 0 0 99 0 0 0 0 105 0 0 0 0 105
,
 0 0 1 0 0 0 0 1 98 0 0 0 0 98 0 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
`G:=sub<GL(4,GF(113))| [99,0,0,0,0,99,0,0,0,0,105,0,0,0,0,105],[0,0,98,0,0,0,0,98,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112] >;`

C28.76C24 in GAP, Magma, Sage, TeX

`C_{28}._{76}C_2^4`
`% in TeX`

`G:=Group("C28.76C2^4");`
`// GroupNames label`

`G:=SmallGroup(448,1272);`
`// by ID`

`G=gap.SmallGroup(448,1272);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,1123,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^28=c^2=d^2=e^2=1,b^2=a^21,b*a*b^-1=a^13,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^14*b,d*c*d=a^14*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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