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G = C24.27C23order 192 = 26·3

20th non-split extension by C24 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24.27C23
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C4○D24 — C24.27C23
 Lower central C3 — C6 — C12 — C24 — C24.27C23
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C24.27C23
G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a9b, dcd-1=a12c >

Subgroups: 248 in 82 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, M5(2), SD32, Q32, C2×Q16, C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C2×C24, C3×Q16, C3×Q16, C4○D12, C6×Q8, Q32⋊C2, C12.C8, C8.6D6, C3⋊Q32, C4○D24, C6×Q16, C24.27C23
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, Q32⋊C2, C2×D4⋊S3, C24.27C23

Character table of C24.27C23

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 24 2 2 2 8 8 24 2 2 2 2 2 4 4 4 8 8 8 8 12 12 12 12 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ7 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 -2 0 2 -2 2 0 0 0 2 -2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 2 -2 2 -2 orthogonal lifted from D4 ρ10 2 2 -2 0 -1 -2 2 2 -2 0 -1 1 1 2 2 -2 -1 1 -1 1 1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 -2 0 -1 -2 2 -2 2 0 -1 1 1 2 2 -2 -1 1 1 -1 -1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 0 -1 2 2 -2 -2 0 -1 -1 -1 2 2 2 -1 -1 1 1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ13 2 2 2 0 -1 2 2 2 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 0 2 2 2 0 0 0 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ15 2 2 -2 0 2 2 -2 0 0 0 2 -2 -2 0 0 0 -2 2 0 0 0 0 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 0 2 2 -2 0 0 0 2 -2 -2 0 0 0 -2 2 0 0 0 0 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 0 2 -2 -2 0 0 0 2 2 2 0 0 0 -2 -2 0 0 0 0 √2 -√2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ18 2 2 2 0 2 -2 -2 0 0 0 2 2 2 0 0 0 -2 -2 0 0 0 0 -√2 √2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ19 2 2 -2 0 -1 -2 2 0 0 0 -1 1 1 -2 -2 2 -1 1 √-3 √-3 -√-3 -√-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ20 2 2 2 0 -1 2 2 0 0 0 -1 -1 -1 -2 -2 -2 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 -2 0 -1 -2 2 0 0 0 -1 1 1 -2 -2 2 -1 1 -√-3 -√-3 √-3 √-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ22 2 2 2 0 -1 2 2 0 0 0 -1 -1 -1 -2 -2 -2 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ23 4 4 4 0 -2 -4 -4 0 0 0 -2 -2 -2 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 -4 0 -2 4 -4 0 0 0 -2 2 2 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 -4 0 0 4 0 0 0 0 0 -4 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 symplectic lifted from Q32⋊C2, Schur index 2 ρ26 4 -4 0 0 4 0 0 0 0 0 -4 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 symplectic lifted from Q32⋊C2, Schur index 2 ρ27 4 -4 0 0 -2 0 0 0 0 0 2 -2√-3 2√-3 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 √-6 √2 -√-6 -√2 complex faithful ρ28 4 -4 0 0 -2 0 0 0 0 0 2 -2√-3 2√-3 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 -√-6 -√2 √-6 √2 complex faithful ρ29 4 -4 0 0 -2 0 0 0 0 0 2 2√-3 -2√-3 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 √-6 -√2 -√-6 √2 complex faithful ρ30 4 -4 0 0 -2 0 0 0 0 0 2 2√-3 -2√-3 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 -√-6 √2 √-6 -√2 complex faithful

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

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

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

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

Matrix representation of C24.27C23 in GL4(𝔽7) generated by

 0 2 2 2 1 0 5 3 0 4 5 5 5 3 6 6
,
 1 6 1 0 0 6 2 0 0 0 1 0 1 3 5 6
,
 2 4 0 3 0 3 1 6 3 2 5 0 3 5 1 4
,
 4 3 0 1 2 0 5 2 2 2 6 5 5 2 6 4
`G:=sub<GL(4,GF(7))| [0,1,0,5,2,0,4,3,2,5,5,6,2,3,5,6],[1,0,0,1,6,6,0,3,1,2,1,5,0,0,0,6],[2,0,3,3,4,3,2,5,0,1,5,1,3,6,0,4],[4,2,2,5,3,0,2,2,0,5,6,6,1,2,5,4] >;`

C24.27C23 in GAP, Magma, Sage, TeX

`C_{24}._{27}C_2^3`
`% in TeX`

`G:=Group("C24.27C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,738);`
`// by ID`

`G=gap.SmallGroup(192,738);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,184,675,185,192,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=a^12*c>;`
`// generators/relations`

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