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## G = C28.23D4order 224 = 25·7

### 23rd non-split extension by C28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.23D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×D28 — C28.23D4
 Lower central C7 — C2×C14 — C28.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C28.23D4
G = < a,b,c | a28=b4=c2=1, bab-1=a13, cac=a-1, cbc=a14b-1 >

Subgroups: 350 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4.4D4, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C4×Dic7, D14⋊C4, C2×D28, Q8×C14, C28.23D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C7⋊D4, C22×D7, Q82D7, C2×C7⋊D4, C28.23D4

Smallest permutation representation of C28.23D4
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 73 42 91)(2 58 43 104)(3 71 44 89)(4 84 45 102)(5 69 46 87)(6 82 47 100)(7 67 48 85)(8 80 49 98)(9 65 50 111)(10 78 51 96)(11 63 52 109)(12 76 53 94)(13 61 54 107)(14 74 55 92)(15 59 56 105)(16 72 29 90)(17 57 30 103)(18 70 31 88)(19 83 32 101)(20 68 33 86)(21 81 34 99)(22 66 35 112)(23 79 36 97)(24 64 37 110)(25 77 38 95)(26 62 39 108)(27 75 40 93)(28 60 41 106)
(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 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14I 28A ··· 28R order 1 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 28 28 2 2 4 4 14 14 14 14 2 2 2 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 D7 C4○D4 D14 C7⋊D4 Q8⋊2D7 kernel C28.23D4 C4×Dic7 D14⋊C4 C2×D28 Q8×C14 C28 C2×Q8 C14 C2×C4 C4 C2 # reps 1 1 4 1 1 2 3 4 9 12 6

Matrix representation of C28.23D4 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 22 26 0 0 0 0 10 0 0 0 0 0 0 0 12 12 0 0 0 0 0 17
,
 6 19 0 0 0 0 24 23 0 0 0 0 0 0 0 3 0 0 0 0 10 0 0 0 0 0 0 0 1 1 0 0 0 0 0 28
,
 1 0 0 0 0 0 7 28 0 0 0 0 0 0 0 26 0 0 0 0 19 0 0 0 0 0 0 0 1 0 0 0 0 0 27 28

`G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,10,0,0,0,0,26,0,0,0,0,0,0,0,12,0,0,0,0,0,12,17],[6,24,0,0,0,0,19,23,0,0,0,0,0,0,0,10,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,28],[1,7,0,0,0,0,0,28,0,0,0,0,0,0,0,19,0,0,0,0,26,0,0,0,0,0,0,0,1,27,0,0,0,0,0,28] >;`

C28.23D4 in GAP, Magma, Sage, TeX

`C_{28}._{23}D_4`
`% in TeX`

`G:=Group("C28.23D4");`
`// GroupNames label`

`G:=SmallGroup(224,142);`
`// by ID`

`G=gap.SmallGroup(224,142);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,188,86,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=a^14*b^-1>;`
`// generators/relations`

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