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## G = D12.33C23order 192 = 26·3

### 14th non-split extension by D12 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12.33C23
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — Q8○D12 — D12.33C23
 Lower central C3 — C6 — C12 — D12.33C23
 Upper central C1 — C2 — C4○D4 — 2+ 1+4

Generators and relations for D12.33C23
G = < a,b,c,d,e | a12=b2=1, c2=d2=e2=a6, bab=a-1, ac=ca, ad=da, eae-1=a7, bc=cb, bd=db, ebe-1=a3b, dcd-1=a6c, ce=ec, de=ed >

Subgroups: 600 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3, C6, C6 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×6], D4 [×10], Q8 [×2], Q8 [×6], C23 [×3], Dic3 [×3], C12, C12 [×3], C12, D6, C2×C6 [×3], C2×C6 [×6], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×3], C4○D4 [×7], C3⋊C8, C3⋊C8 [×3], Dic6 [×3], Dic6 [×3], C4×S3 [×3], D12, C2×Dic3 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×3], C3×D4 [×6], C3×D4 [×6], C3×Q8 [×2], C22×C6 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3 [×3], D4.S3 [×9], Q82S3, C3⋊Q16 [×3], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×3], S3×Q8, C6×D4 [×3], C6×D4 [×3], C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4, D4○SD16, D126C22 [×3], C2×D4.S3 [×3], D4.Dic3, Q8.13D6 [×3], Q8.14D6 [×3], Q8○D12, C3×2+ 1+4, D12.33C23
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, D4○SD16, C22×C3⋊D4, D12.33C23

Smallest permutation representation of D12.33C23
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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 37 31 43)(26 38 32 44)(27 39 33 45)(28 40 34 46)(29 41 35 47)(30 42 36 48)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 13 7 19)(2 20 8 14)(3 15 9 21)(4 22 10 16)(5 17 11 23)(6 24 12 18)(25 46 31 40)(26 41 32 47)(27 48 33 42)(28 43 34 37)(29 38 35 44)(30 45 36 39)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 D4○SD16 D12.33C23 kernel D12.33C23 D12⋊6C22 C2×D4.S3 D4.Dic3 Q8.13D6 Q8.14D6 Q8○D12 C3×2+ 1+4 2+ 1+4 C3×D4 C3×Q8 C2×D4 C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 3 3 1 1 1 3 1 3 4 6 2 2 1

Matrix representation of D12.33C23 in GL6(𝔽73)

 64 0 0 0 0 0 71 8 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 72 1 72 2 0 0 0 1 72 1
,
 55 7 0 0 0 0 6 18 0 0 0 0 0 0 6 67 0 61 0 0 67 6 61 12 0 0 6 6 0 0 0 0 12 67 6 61
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 1 72 1 71 0 0 1 0 0 0 0 0 0 1 72 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 72 1 72 2 0 0 72 0 72 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 1 72 1 71 0 0 72 0 0 0 0 0 72 1 0 1

`G:=sub<GL(6,GF(73))| [64,71,0,0,0,0,0,8,0,0,0,0,0,0,0,72,72,0,0,0,1,0,1,1,0,0,0,0,72,72,0,0,0,0,2,1],[55,6,0,0,0,0,7,18,0,0,0,0,0,0,6,67,6,12,0,0,67,6,6,67,0,0,0,61,0,6,0,0,61,12,0,61],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,1,0,0,72,1,0,72,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,72,0,1,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,72,72,0,0,0,72,0,1,0,0,1,1,0,0,0,0,0,71,0,1] >;`

D12.33C23 in GAP, Magma, Sage, TeX

`D_{12}._{33}C_2^3`
`% in TeX`

`G:=Group("D12.33C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,1395);`
`// by ID`

`G=gap.SmallGroup(192,1395);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,136,1684,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^2=1,c^2=d^2=e^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^7,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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