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## G = C28.70C24order 448 = 26·7

### 17th non-split extension by C28 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C28.70C24
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×C4○D28 — C28.70C24
 Lower central C7 — C14 — C28.70C24
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C28.70C24
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a41, ac=ca, dad=a29, cbc=a28b, bd=db, cd=dc >

Subgroups: 932 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, Q8○M4(2), C8×D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D28.2C4, D7×M4(2), D28.C4, C2×C4.Dic7, C14×M4(2), C2×C4○D28, C28.70C24
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C23×C4, C4×D7, C22×D7, Q8○M4(2), C2×C4×D7, C23×D7, D7×C22×C4, C28.70C24

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

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

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

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

94 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 14 14 14 14 1 1 2 2 2 14 14 14 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

94 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 D14 D14 D14 C4×D7 C4×D7 Q8○M4(2) C28.70C24 kernel C28.70C24 D28.2C4 D7×M4(2) D28.C4 C2×C4.Dic7 C14×M4(2) C2×C4○D28 C2×Dic14 C2×D28 C4○D28 C2×C7⋊D4 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C7 C1 # reps 1 4 4 4 1 1 1 2 2 8 4 3 6 12 3 18 6 2 12

Matrix representation of C28.70C24 in GL4(𝔽113) generated by

 104 1 15 92 41 33 43 0 97 0 0 112 31 97 1 89
,
 0 89 0 0 80 0 0 0 0 0 1 0 0 0 24 112
,
 79 7 0 0 61 34 0 0 0 0 29 7 0 0 106 84
,
 1 0 0 0 0 1 0 0 27 30 112 0 83 0 0 112
`G:=sub<GL(4,GF(113))| [104,41,97,31,1,33,0,97,15,43,0,1,92,0,112,89],[0,80,0,0,89,0,0,0,0,0,1,24,0,0,0,112],[79,61,0,0,7,34,0,0,0,0,29,106,0,0,7,84],[1,0,27,83,0,1,30,0,0,0,112,0,0,0,0,112] >;`

C28.70C24 in GAP, Magma, Sage, TeX

`C_{28}._{70}C_2^4`
`% in TeX`

`G:=Group("C28.70C2^4");`
`// GroupNames label`

`G:=SmallGroup(448,1198);`
`// by ID`

`G=gap.SmallGroup(448,1198);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,570,80,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^56=b^2=c^2=d^2=1,b*a*b=a^41,a*c=c*a,d*a*d=a^29,c*b*c=a^28*b,b*d=d*b,c*d=d*c>;`
`// generators/relations`

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