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## G = C24.44D6order 192 = 26·3

### 33rd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.44D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C24.44D6
 Lower central C3 — C2×C6 — C24.44D6
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.44D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 928 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×4], C22 [×25], S3 [×3], C6 [×3], C6 [×6], C2×C4 [×3], C2×C4 [×16], D4 [×18], Q8 [×2], C23 [×4], C23 [×12], Dic3 [×2], Dic3 [×5], C12 [×3], D6 [×2], D6 [×9], C2×C6, C2×C6 [×4], C2×C6 [×14], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24, C24, Dic6 [×2], C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×9], C2×C12 [×3], C3×D4 [×5], C22×S3 [×2], C22×S3 [×5], C22×C6 [×4], C22×C6 [×5], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×5], C3×C22⋊C4 [×3], C2×Dic6, S3×C2×C4 [×2], D42S3 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×6], C2×C3⋊D4 [×4], C6×D4 [×3], S3×C23, C23×C6, D45D4, Dic3.D4, S3×C22⋊C4, Dic34D4, C23.9D6, C23.11D6, C23.21D6, D4×Dic3, C232D6, D63D4, C23.14D6 [×2], C2×C6.D4, C3×C22≀C2, C2×D42S3, C22×C3⋊D4, C24.44D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], D42S3 [×2], S3×C23, D45D4, C2×S3×D4, C2×D42S3, D46D6, C24.44D6

Smallest permutation representation of C24.44D6
On 48 points
Generators in S48
```(1 7)(2 37)(3 9)(4 39)(5 11)(6 41)(8 43)(10 45)(12 47)(13 26)(14 20)(15 28)(16 22)(17 30)(18 24)(19 32)(21 34)(23 36)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)
(1 14)(2 21)(3 16)(4 23)(5 18)(6 13)(7 20)(8 15)(9 22)(10 17)(11 24)(12 19)(25 46)(26 41)(27 48)(28 43)(29 38)(30 45)(31 40)(32 47)(33 42)(34 37)(35 44)(36 39)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 12 7 6)(2 5 8 11)(3 10 9 4)(13 33 19 27)(14 26 20 32)(15 31 21 25)(16 36 22 30)(17 29 23 35)(18 34 24 28)(37 40 43 46)(38 45 44 39)(41 48 47 42)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 D4⋊2S3 D4⋊6D6 kernel C24.44D6 Dic3.D4 S3×C22⋊C4 Dic3⋊4D4 C23.9D6 C23.11D6 C23.21D6 D4×Dic3 C23⋊2D6 D6⋊3D4 C23.14D6 C2×C6.D4 C3×C22≀C2 C2×D4⋊2S3 C22×C3⋊D4 C22≀C2 C3⋊D4 C22⋊C4 C2×D4 C24 C2×C6 C6 C22 C22 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 4 3 3 1 4 1 2 2 2

Matrix representation of C24.44D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 2 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 8 5 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 1 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;`

C24.44D6 in GAP, Magma, Sage, TeX

`C_2^4._{44}D_6`
`% in TeX`

`G:=Group("C2^4.44D6");`
`// GroupNames label`

`G:=SmallGroup(192,1150);`
`// by ID`

`G=gap.SmallGroup(192,1150);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;`
`// generators/relations`

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