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## G = C22⋊(Q8⋊C9)  order 288 = 25·32

### The semidirect product of C22 and Q8⋊C9 acting via Q8⋊C9/C3×Q8=C3

Aliases: C22⋊(Q8⋊C9), (C3×Q8).1A4, (C22×Q8)⋊3C9, C3.(Q8⋊A4), Q81(C3.A4), C6.1(C22⋊A4), (C22×C6).10A4, C2.1(C24⋊C9), C23.5(C3.A4), (C2×C6).3SL2(𝔽3), (Q8×C2×C6).3C3, SmallGroup(288,350)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — C22⋊(Q8⋊C9)
 Chief series C1 — C2 — Q8 — C22×Q8 — Q8×C2×C6 — C22⋊(Q8⋊C9)
 Lower central C22×Q8 — C22⋊(Q8⋊C9)
 Upper central C1 — C6

Generators and relations for C22⋊(Q8⋊C9)
G = < a,b,c,d,e | a2=b2=c4=e9=1, d2=c2, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 213 in 73 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, Q8, Q8, C23, C9, C12, C2×C6, C2×C6, C22×C4, C2×Q8, C18, C2×C12, C3×Q8, C3×Q8, C22×C6, C22×Q8, C3.A4, C22×C12, C6×Q8, Q8⋊C9, C2×C3.A4, Q8×C2×C6, C22⋊(Q8⋊C9)
Quotients: C1, C3, C9, A4, SL2(𝔽3), C3.A4, C22⋊A4, Q8⋊C9, Q8⋊A4, C24⋊C9, C22⋊(Q8⋊C9)

Smallest permutation representation of C22⋊(Q8⋊C9)
On 72 points
Generators in S72
```(1 46)(2 47)(4 49)(5 50)(7 52)(8 53)(10 21)(12 23)(13 24)(15 26)(16 27)(18 20)(29 64)(30 65)(32 67)(33 68)(35 70)(36 71)(37 57)(39 59)(40 60)(42 62)(43 63)(45 56)
(2 47)(3 48)(5 50)(6 51)(8 53)(9 54)(10 21)(11 22)(13 24)(14 25)(16 27)(17 19)(28 72)(30 65)(31 66)(33 68)(34 69)(36 71)(37 57)(38 58)(40 60)(41 61)(43 63)(44 55)
(1 18 46 20)(2 40 47 60)(3 31 48 66)(4 12 49 23)(5 43 50 63)(6 34 51 69)(7 15 52 26)(8 37 53 57)(9 28 54 72)(10 30 21 65)(11 61 22 41)(13 33 24 68)(14 55 25 44)(16 36 27 71)(17 58 19 38)(29 39 64 59)(32 42 67 62)(35 45 70 56)
(1 39 46 59)(2 30 47 65)(3 11 48 22)(4 42 49 62)(5 33 50 68)(6 14 51 25)(7 45 52 56)(8 36 53 71)(9 17 54 19)(10 60 21 40)(12 32 23 67)(13 63 24 43)(15 35 26 70)(16 57 27 37)(18 29 20 64)(28 38 72 58)(31 41 66 61)(34 44 69 55)
(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)```

`G:=sub<Sym(72)| (1,46)(2,47)(4,49)(5,50)(7,52)(8,53)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(29,64)(30,65)(32,67)(33,68)(35,70)(36,71)(37,57)(39,59)(40,60)(42,62)(43,63)(45,56), (2,47)(3,48)(5,50)(6,51)(8,53)(9,54)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,72)(30,65)(31,66)(33,68)(34,69)(36,71)(37,57)(38,58)(40,60)(41,61)(43,63)(44,55), (1,18,46,20)(2,40,47,60)(3,31,48,66)(4,12,49,23)(5,43,50,63)(6,34,51,69)(7,15,52,26)(8,37,53,57)(9,28,54,72)(10,30,21,65)(11,61,22,41)(13,33,24,68)(14,55,25,44)(16,36,27,71)(17,58,19,38)(29,39,64,59)(32,42,67,62)(35,45,70,56), (1,39,46,59)(2,30,47,65)(3,11,48,22)(4,42,49,62)(5,33,50,68)(6,14,51,25)(7,45,52,56)(8,36,53,71)(9,17,54,19)(10,60,21,40)(12,32,23,67)(13,63,24,43)(15,35,26,70)(16,57,27,37)(18,29,20,64)(28,38,72,58)(31,41,66,61)(34,44,69,55), (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)>;`

`G:=Group( (1,46)(2,47)(4,49)(5,50)(7,52)(8,53)(10,21)(12,23)(13,24)(15,26)(16,27)(18,20)(29,64)(30,65)(32,67)(33,68)(35,70)(36,71)(37,57)(39,59)(40,60)(42,62)(43,63)(45,56), (2,47)(3,48)(5,50)(6,51)(8,53)(9,54)(10,21)(11,22)(13,24)(14,25)(16,27)(17,19)(28,72)(30,65)(31,66)(33,68)(34,69)(36,71)(37,57)(38,58)(40,60)(41,61)(43,63)(44,55), (1,18,46,20)(2,40,47,60)(3,31,48,66)(4,12,49,23)(5,43,50,63)(6,34,51,69)(7,15,52,26)(8,37,53,57)(9,28,54,72)(10,30,21,65)(11,61,22,41)(13,33,24,68)(14,55,25,44)(16,36,27,71)(17,58,19,38)(29,39,64,59)(32,42,67,62)(35,45,70,56), (1,39,46,59)(2,30,47,65)(3,11,48,22)(4,42,49,62)(5,33,50,68)(6,14,51,25)(7,45,52,56)(8,36,53,71)(9,17,54,19)(10,60,21,40)(12,32,23,67)(13,63,24,43)(15,35,26,70)(16,57,27,37)(18,29,20,64)(28,38,72,58)(31,41,66,61)(34,44,69,55), (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) );`

`G=PermutationGroup([[(1,46),(2,47),(4,49),(5,50),(7,52),(8,53),(10,21),(12,23),(13,24),(15,26),(16,27),(18,20),(29,64),(30,65),(32,67),(33,68),(35,70),(36,71),(37,57),(39,59),(40,60),(42,62),(43,63),(45,56)], [(2,47),(3,48),(5,50),(6,51),(8,53),(9,54),(10,21),(11,22),(13,24),(14,25),(16,27),(17,19),(28,72),(30,65),(31,66),(33,68),(34,69),(36,71),(37,57),(38,58),(40,60),(41,61),(43,63),(44,55)], [(1,18,46,20),(2,40,47,60),(3,31,48,66),(4,12,49,23),(5,43,50,63),(6,34,51,69),(7,15,52,26),(8,37,53,57),(9,28,54,72),(10,30,21,65),(11,61,22,41),(13,33,24,68),(14,55,25,44),(16,36,27,71),(17,58,19,38),(29,39,64,59),(32,42,67,62),(35,45,70,56)], [(1,39,46,59),(2,30,47,65),(3,11,48,22),(4,42,49,62),(5,33,50,68),(6,14,51,25),(7,45,52,56),(8,36,53,71),(9,17,54,19),(10,60,21,40),(12,32,23,67),(13,63,24,43),(15,35,26,70),(16,57,27,37),(18,29,20,64),(28,38,72,58),(31,41,66,61),(34,44,69,55)], [(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)]])`

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A ··· 12H 18A ··· 18F order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 3 3 1 1 6 6 6 6 1 1 3 3 3 3 16 ··· 16 6 ··· 6 16 ··· 16

36 irreducible representations

 dim 1 1 1 2 2 2 3 3 3 3 6 6 type + - + + - image C1 C3 C9 SL2(𝔽3) SL2(𝔽3) Q8⋊C9 A4 A4 C3.A4 C3.A4 Q8⋊A4 C22⋊(Q8⋊C9) kernel C22⋊(Q8⋊C9) Q8×C2×C6 C22×Q8 C2×C6 C2×C6 C22 C3×Q8 C22×C6 Q8 C23 C3 C1 # reps 1 2 6 1 2 6 4 1 8 2 1 2

Matrix representation of C22⋊(Q8⋊C9) in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 36 0 1 0 0 36 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 36 0 0 0 1 36 0
,
 27 26 0 0 0 26 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 26 10 0 0 0 0 0 0 0 9 0 0 9 0 0 0 0 0 9 0

`G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[27,26,0,0,0,26,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,26,0,0,0,0,10,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,9,0,0] >;`

C22⋊(Q8⋊C9) in GAP, Magma, Sage, TeX

`C_2^2\rtimes (Q_8\rtimes C_9)`
`% in TeX`

`G:=Group("C2^2:(Q8:C9)");`
`// GroupNames label`

`G:=SmallGroup(288,350);`
`// by ID`

`G=gap.SmallGroup(288,350);`
`# by ID`

`G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,21,380,759,2524,172,4541,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^9=1,d^2=c^2,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;`
`// generators/relations`

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