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G = C12.C24order 192 = 26·3

35th non-split extension by C12 of C24 acting via C24/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.C24
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C12.C24
 Lower central C3 — C6 — C12 — C12.C24
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for C12.C24
G = < a,b,c,d,e | a12=b2=c2=e2=1, d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a9b, cd=dc, ece=a6c, de=ed >

Subgroups: 616 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, D8⋊C22, C2×C4.Dic3, D126C22, Q8.11D6, D4⋊D6, Q8.13D6, Q8.14D6, C2×C4○D12, C6×C4○D4, C12.C24
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C3⋊D4, S3×C23, D8⋊C22, C22×C3⋊D4, C12.C24

Smallest permutation representation of C12.C24
On 48 points
Generators in S48
```(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)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(26 36)(27 35)(28 34)(29 33)(30 32)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 37)(26 44)(27 39)(28 46)(29 41)(30 48)(31 43)(32 38)(33 45)(34 40)(35 47)(36 42)```

`G:=sub<Sym(48)| (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), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42)>;`

`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), (2,12)(3,11)(4,10)(5,9)(6,8)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(26,36)(27,35)(28,34)(29,33)(30,32)(37,46)(38,45)(39,44)(40,43)(41,42)(47,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,43,19,37)(14,44,20,38)(15,45,21,39)(16,46,22,40)(17,47,23,41)(18,48,24,42), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42) );`

`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)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(26,36),(27,35),(28,34),(29,33),(30,32),(37,46),(38,45),(39,44),(40,43),(41,42),(47,48)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,43,19,37),(14,44,20,38),(15,45,21,39),(16,46,22,40),(17,47,23,41),(18,48,24,42)], [(1,13),(2,20),(3,15),(4,22),(5,17),(6,24),(7,19),(8,14),(9,21),(10,16),(11,23),(12,18),(25,37),(26,44),(27,39),(28,46),(29,41),(30,48),(31,43),(32,38),(33,45),(34,40),(35,47),(36,42)]])`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D ··· 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 2 2 2 4 4 12 12 2 1 1 2 2 2 4 4 12 12 2 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 D8⋊C22 C12.C24 kernel C12.C24 C2×C4.Dic3 D12⋊6C22 Q8.11D6 D4⋊D6 Q8.13D6 Q8.14D6 C2×C4○D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C3 C1 # reps 1 1 2 2 2 4 2 1 1 1 3 1 1 1 1 4 6 2 2 4

Matrix representation of C12.C24 in GL4(𝔽73) generated by

 14 7 0 0 66 7 0 0 0 0 59 66 0 0 7 66
,
 0 72 0 0 72 0 0 0 0 0 59 66 0 0 7 14
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 46 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
`G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,59,7,0,0,66,66],[0,72,0,0,72,0,0,0,0,0,59,7,0,0,66,14],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;`

C12.C24 in GAP, Magma, Sage, TeX

`C_{12}.C_2^4`
`% in TeX`

`G:=Group("C12.C2^4");`
`// GroupNames label`

`G:=SmallGroup(192,1381);`
`// by ID`

`G=gap.SmallGroup(192,1381);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,1684,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^2=c^2=e^2=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^7,b*c=c*b,b*d=d*b,e*b*e=a^9*b,c*d=d*c,e*c*e=a^6*c,d*e=e*d>;`
`// generators/relations`

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