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## G = C28.A4order 336 = 24·3·7

### The non-split extension by C28 of A4 acting via A4/C22=C3

Aliases: C28.A4, C4.(C7⋊A4), C72(C4.A4), C14.A43C2, C14.6(C2×A4), (C7×Q8).6C6, C4○D4⋊(C7⋊C3), Q8.(C2×C7⋊C3), C2.3(C2×C7⋊A4), (C7×C4○D4)⋊3C3, SmallGroup(336,173)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×Q8 — C28.A4
 Chief series C1 — C2 — C14 — C7×Q8 — C14.A4 — C28.A4
 Lower central C7×Q8 — C28.A4
 Upper central C1 — C4

Generators and relations for C28.A4
G = < a,b,c,d | a28=d3=1, b2=c2=a14, ab=ba, ac=ca, dad-1=a25, cbc-1=a14b, dbd-1=a14bc, dcd-1=b >

Smallest permutation representation of C28.A4
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 31 15 45)(2 32 16 46)(3 33 17 47)(4 34 18 48)(5 35 19 49)(6 36 20 50)(7 37 21 51)(8 38 22 52)(9 39 23 53)(10 40 24 54)(11 41 25 55)(12 42 26 56)(13 43 27 29)(14 44 28 30)(57 111 71 97)(58 112 72 98)(59 85 73 99)(60 86 74 100)(61 87 75 101)(62 88 76 102)(63 89 77 103)(64 90 78 104)(65 91 79 105)(66 92 80 106)(67 93 81 107)(68 94 82 108)(69 95 83 109)(70 96 84 110)
(1 65 15 79)(2 66 16 80)(3 67 17 81)(4 68 18 82)(5 69 19 83)(6 70 20 84)(7 71 21 57)(8 72 22 58)(9 73 23 59)(10 74 24 60)(11 75 25 61)(12 76 26 62)(13 77 27 63)(14 78 28 64)(29 103 43 89)(30 104 44 90)(31 105 45 91)(32 106 46 92)(33 107 47 93)(34 108 48 94)(35 109 49 95)(36 110 50 96)(37 111 51 97)(38 112 52 98)(39 85 53 99)(40 86 54 100)(41 87 55 101)(42 88 56 102)
(2 10 26)(3 19 23)(4 28 20)(5 9 17)(6 18 14)(7 27 11)(12 16 24)(13 25 21)(29 75 97)(30 84 94)(31 65 91)(32 74 88)(33 83 85)(34 64 110)(35 73 107)(36 82 104)(37 63 101)(38 72 98)(39 81 95)(40 62 92)(41 71 89)(42 80 86)(43 61 111)(44 70 108)(45 79 105)(46 60 102)(47 69 99)(48 78 96)(49 59 93)(50 68 90)(51 77 87)(52 58 112)(53 67 109)(54 76 106)(55 57 103)(56 66 100)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 7A 7B 12A 12B 12C 12D 14A 14B 14C ··· 14H 28A 28B 28C 28D 28E ··· 28J order 1 2 2 3 3 4 4 4 6 6 7 7 12 12 12 12 14 14 14 ··· 14 28 28 28 28 28 ··· 28 size 1 1 6 28 28 1 1 6 28 28 3 3 28 28 28 28 3 3 6 ··· 6 3 3 3 3 6 ··· 6

34 irreducible representations

 dim 1 1 1 1 2 3 3 3 3 3 3 6 type + + + + image C1 C2 C3 C6 C4.A4 A4 C7⋊C3 C2×A4 C2×C7⋊C3 C7⋊A4 C2×C7⋊A4 C28.A4 kernel C28.A4 C14.A4 C7×C4○D4 C7×Q8 C7 C28 C4○D4 C14 Q8 C4 C2 C1 # reps 1 1 2 2 6 1 2 1 2 6 6 4

Matrix representation of C28.A4 in GL5(𝔽337)

 189 0 0 0 0 0 189 0 0 0 0 0 0 0 1 0 0 1 213 212 0 0 212 336 336
,
 128 129 0 0 0 129 209 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 336 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 209 208 0 0 0 0 0 1 0 0 0 0 212 336 336 0 0 0 1 0

`G:=sub<GL(5,GF(337))| [189,0,0,0,0,0,189,0,0,0,0,0,0,1,212,0,0,0,213,336,0,0,1,212,336],[128,129,0,0,0,129,209,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,209,0,0,0,0,208,0,0,0,0,0,1,212,0,0,0,0,336,1,0,0,0,336,0] >;`

C28.A4 in GAP, Magma, Sage, TeX

`C_{28}.A_4`
`% in TeX`

`G:=Group("C28.A4");`
`// GroupNames label`

`G:=SmallGroup(336,173);`
`// by ID`

`G=gap.SmallGroup(336,173);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-7,-2,1008,116,518,225,735,357,730]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=d^3=1,b^2=c^2=a^14,a*b=b*a,a*c=c*a,d*a*d^-1=a^25,c*b*c^-1=a^14*b,d*b*d^-1=a^14*b*c,d*c*d^-1=b>;`
`// generators/relations`

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