Copied to
clipboard

## G = C24.78C23order 192 = 26·3

### 24th non-split extension by C24 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C24.78C23
 Chief series C1 — C3 — C6 — C12 — C24 — C3⋊C16 — C2×C3⋊C16 — C24.78C23
 Lower central C3 — C6 — C24.78C23
 Upper central C1 — C8 — C8○D4

Generators and relations for C24.78C23
G = < a,b,c,d | a24=c2=d2=1, b2=a9, bab-1=a17, ac=ca, ad=da, bc=cb, bd=db, dcd=a12c >

Subgroups: 112 in 84 conjugacy classes, 67 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C16, C2×C8, M4(2), C4○D4, C24, C24, C2×C12, C3×D4, C3×Q8, C2×C16, M5(2), C8○D4, C3⋊C16, C3⋊C16, C2×C24, C3×M4(2), C3×C4○D4, D4○C16, C2×C3⋊C16, C12.C8, C3×C8○D4, C24.78C23
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, C22×C4, C3⋊C8, C2×Dic3, C22×S3, C22×C8, C2×C3⋊C8, C22×Dic3, D4○C16, C22×C3⋊C8, C24.78C23

Smallest permutation representation of C24.78C23
On 96 points
Generators in S96
```(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)
(1 55 10 64 19 49 4 58 13 67 22 52 7 61 16 70)(2 72 11 57 20 66 5 51 14 60 23 69 8 54 17 63)(3 65 12 50 21 59 6 68 15 53 24 62 9 71 18 56)(25 93 34 78 43 87 28 96 37 81 46 90 31 75 40 84)(26 86 35 95 44 80 29 89 38 74 47 83 32 92 41 77)(27 79 36 88 45 73 30 82 39 91 48 76 33 85 42 94)
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 46)(2 47)(3 48)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 8E ··· 8J 12A 12B 12C 12D 12E 16A ··· 16H 16I ··· 16T 24A 24B 24C 24D 24E ··· 24J order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 6 8 8 8 8 8 ··· 8 12 12 12 12 12 16 ··· 16 16 ··· 16 24 24 24 24 24 ··· 24 size 1 1 2 2 2 2 1 1 2 2 2 2 4 4 4 1 1 1 1 2 ··· 2 2 2 4 4 4 3 ··· 3 6 ··· 6 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 C8 C8 S3 D6 Dic3 Dic3 C3⋊C8 C3⋊C8 D4○C16 C24.78C23 kernel C24.78C23 C2×C3⋊C16 C12.C8 C3×C8○D4 C3×M4(2) C3×C4○D4 C3×D4 C3×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 6 2 12 4 1 3 3 1 6 2 8 4

Matrix representation of C24.78C23 in GL4(𝔽97) generated by

 47 0 0 0 0 47 0 0 0 0 75 75 0 0 22 0
,
 85 0 0 0 0 85 0 0 0 0 87 18 0 0 28 10
,
 1 0 0 0 1 96 0 0 0 0 1 0 0 0 0 1
,
 1 95 0 0 0 96 0 0 0 0 96 0 0 0 0 96
`G:=sub<GL(4,GF(97))| [47,0,0,0,0,47,0,0,0,0,75,22,0,0,75,0],[85,0,0,0,0,85,0,0,0,0,87,28,0,0,18,10],[1,1,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,95,96,0,0,0,0,96,0,0,0,0,96] >;`

C24.78C23 in GAP, Magma, Sage, TeX

`C_{24}._{78}C_2^3`
`% in TeX`

`G:=Group("C24.78C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,699);`
`// by ID`

`G=gap.SmallGroup(192,699);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,80,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=c^2=d^2=1,b^2=a^9,b*a*b^-1=a^17,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^12*c>;`
`// generators/relations`

׿
×
𝔽