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## G = C28.C23order 224 = 25·7

### 15th non-split extension by C28 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.C23
 Chief series C1 — C7 — C14 — C28 — D28 — C4○D28 — C28.C23
 Lower central C7 — C14 — C28 — C28.C23
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C28.C23
G = < a,b,c,d | a28=b2=1, c2=d2=a14, bab=a-1, ac=ca, dad-1=a15, bc=cb, dbd-1=a21b, dcd-1=a14c >

Subgroups: 222 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, D7, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C8.C22, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, C4.Dic7, Q8⋊D7, C7⋊Q16, C4○D28, Q8×C14, C28.C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C7⋊D4, C22×D7, C2×C7⋊D4, C28.C23

Smallest permutation representation of C28.C23
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 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 36)(30 35)(31 34)(32 33)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(107 112)(108 111)(109 110)
(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)(29 106 43 92)(30 107 44 93)(31 108 45 94)(32 109 46 95)(33 110 47 96)(34 111 48 97)(35 112 49 98)(36 85 50 99)(37 86 51 100)(38 87 52 101)(39 88 53 102)(40 89 54 103)(41 90 55 104)(42 91 56 105)
(1 36 15 50)(2 51 16 37)(3 38 17 52)(4 53 18 39)(5 40 19 54)(6 55 20 41)(7 42 21 56)(8 29 22 43)(9 44 23 30)(10 31 24 45)(11 46 25 32)(12 33 26 47)(13 48 27 34)(14 35 28 49)(57 91 71 105)(58 106 72 92)(59 93 73 107)(60 108 74 94)(61 95 75 109)(62 110 76 96)(63 97 77 111)(64 112 78 98)(65 99 79 85)(66 86 80 100)(67 101 81 87)(68 88 82 102)(69 103 83 89)(70 90 84 104)```

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 14A ··· 14I 28A ··· 28R order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 14 ··· 14 28 ··· 28 size 1 1 2 28 2 2 4 4 28 2 2 2 28 28 2 ··· 2 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C7⋊D4 C7⋊D4 C8.C22 C28.C23 kernel C28.C23 C4.Dic7 Q8⋊D7 C7⋊Q16 C4○D28 Q8×C14 C28 C2×C14 C2×Q8 C2×C4 Q8 C4 C22 C7 C1 # reps 1 1 2 2 1 1 1 1 3 3 6 6 6 1 6

Matrix representation of C28.C23 in GL4(𝔽113) generated by

 16 2 111 24 50 27 24 111 22 76 86 111 101 22 63 97
,
 103 89 0 0 103 10 0 0 94 100 112 0 40 100 89 1
,
 19 55 90 81 91 14 19 90 16 37 99 58 75 16 22 94
,
 60 65 16 22 84 89 46 16 38 84 24 48 69 38 29 53
`G:=sub<GL(4,GF(113))| [16,50,22,101,2,27,76,22,111,24,86,63,24,111,111,97],[103,103,94,40,89,10,100,100,0,0,112,89,0,0,0,1],[19,91,16,75,55,14,37,16,90,19,99,22,81,90,58,94],[60,84,38,69,65,89,84,38,16,46,24,29,22,16,48,53] >;`

C28.C23 in GAP, Magma, Sage, TeX

`C_{28}.C_2^3`
`% in TeX`

`G:=Group("C28.C2^3");`
`// GroupNames label`

`G:=SmallGroup(224,137);`
`// by ID`

`G=gap.SmallGroup(224,137);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,86,579,159,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=1,c^2=d^2=a^14,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^21*b,d*c*d^-1=a^14*c>;`
`// generators/relations`

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