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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

non-abelian, soluble

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

C1C2C22×Q8 — C22⋊(Q8⋊C9)
C1C2Q8C22×Q8Q8×C2×C6 — C22⋊(Q8⋊C9)
C22×Q8 — C22⋊(Q8⋊C9)
C1C6

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 [×2], C3, C4 [×4], C22, C22 [×2], C6, C6 [×2], C2×C4 [×6], Q8 [×4], Q8 [×4], C23, C9, C12 [×4], C2×C6, C2×C6 [×2], C22×C4, C2×Q8 [×4], C18, C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×4], C22×C6, C22×Q8, C3.A4, C22×C12, C6×Q8 [×4], Q8⋊C9 [×4], C2×C3.A4, Q8×C2×C6, C22⋊(Q8⋊C9)
Quotients: C1, C3, C9, A4 [×5], SL2(𝔽3), C3.A4 [×5], 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 30)(2 31)(4 33)(5 34)(7 36)(8 28)(10 56)(12 58)(13 59)(15 61)(16 62)(18 55)(19 71)(21 64)(22 65)(24 67)(25 68)(27 70)(37 50)(38 51)(40 53)(41 54)(43 47)(44 48)
(2 31)(3 32)(5 34)(6 35)(8 28)(9 29)(10 56)(11 57)(13 59)(14 60)(16 62)(17 63)(19 71)(20 72)(22 65)(23 66)(25 68)(26 69)(38 51)(39 52)(41 54)(42 46)(44 48)(45 49)
(1 58 30 12)(2 48 31 44)(3 23 32 66)(4 61 33 15)(5 51 34 38)(6 26 35 69)(7 55 36 18)(8 54 28 41)(9 20 29 72)(10 71 56 19)(11 46 57 42)(13 65 59 22)(14 49 60 45)(16 68 62 25)(17 52 63 39)(21 47 64 43)(24 50 67 37)(27 53 70 40)
(1 47 30 43)(2 22 31 65)(3 60 32 14)(4 50 33 37)(5 25 34 68)(6 63 35 17)(7 53 36 40)(8 19 28 71)(9 57 29 11)(10 54 56 41)(12 64 58 21)(13 48 59 44)(15 67 61 24)(16 51 62 38)(18 70 55 27)(20 46 72 42)(23 49 66 45)(26 52 69 39)
(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)| (1,30)(2,31)(4,33)(5,34)(7,36)(8,28)(10,56)(12,58)(13,59)(15,61)(16,62)(18,55)(19,71)(21,64)(22,65)(24,67)(25,68)(27,70)(37,50)(38,51)(40,53)(41,54)(43,47)(44,48), (2,31)(3,32)(5,34)(6,35)(8,28)(9,29)(10,56)(11,57)(13,59)(14,60)(16,62)(17,63)(19,71)(20,72)(22,65)(23,66)(25,68)(26,69)(38,51)(39,52)(41,54)(42,46)(44,48)(45,49), (1,58,30,12)(2,48,31,44)(3,23,32,66)(4,61,33,15)(5,51,34,38)(6,26,35,69)(7,55,36,18)(8,54,28,41)(9,20,29,72)(10,71,56,19)(11,46,57,42)(13,65,59,22)(14,49,60,45)(16,68,62,25)(17,52,63,39)(21,47,64,43)(24,50,67,37)(27,53,70,40), (1,47,30,43)(2,22,31,65)(3,60,32,14)(4,50,33,37)(5,25,34,68)(6,63,35,17)(7,53,36,40)(8,19,28,71)(9,57,29,11)(10,54,56,41)(12,64,58,21)(13,48,59,44)(15,67,61,24)(16,51,62,38)(18,70,55,27)(20,46,72,42)(23,49,66,45)(26,52,69,39), (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( (1,30)(2,31)(4,33)(5,34)(7,36)(8,28)(10,56)(12,58)(13,59)(15,61)(16,62)(18,55)(19,71)(21,64)(22,65)(24,67)(25,68)(27,70)(37,50)(38,51)(40,53)(41,54)(43,47)(44,48), (2,31)(3,32)(5,34)(6,35)(8,28)(9,29)(10,56)(11,57)(13,59)(14,60)(16,62)(17,63)(19,71)(20,72)(22,65)(23,66)(25,68)(26,69)(38,51)(39,52)(41,54)(42,46)(44,48)(45,49), (1,58,30,12)(2,48,31,44)(3,23,32,66)(4,61,33,15)(5,51,34,38)(6,26,35,69)(7,55,36,18)(8,54,28,41)(9,20,29,72)(10,71,56,19)(11,46,57,42)(13,65,59,22)(14,49,60,45)(16,68,62,25)(17,52,63,39)(21,47,64,43)(24,50,67,37)(27,53,70,40), (1,47,30,43)(2,22,31,65)(3,60,32,14)(4,50,33,37)(5,25,34,68)(6,63,35,17)(7,53,36,40)(8,19,28,71)(9,57,29,11)(10,54,56,41)(12,64,58,21)(13,48,59,44)(15,67,61,24)(16,51,62,38)(18,70,55,27)(20,46,72,42)(23,49,66,45)(26,52,69,39), (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([(1,30),(2,31),(4,33),(5,34),(7,36),(8,28),(10,56),(12,58),(13,59),(15,61),(16,62),(18,55),(19,71),(21,64),(22,65),(24,67),(25,68),(27,70),(37,50),(38,51),(40,53),(41,54),(43,47),(44,48)], [(2,31),(3,32),(5,34),(6,35),(8,28),(9,29),(10,56),(11,57),(13,59),(14,60),(16,62),(17,63),(19,71),(20,72),(22,65),(23,66),(25,68),(26,69),(38,51),(39,52),(41,54),(42,46),(44,48),(45,49)], [(1,58,30,12),(2,48,31,44),(3,23,32,66),(4,61,33,15),(5,51,34,38),(6,26,35,69),(7,55,36,18),(8,54,28,41),(9,20,29,72),(10,71,56,19),(11,46,57,42),(13,65,59,22),(14,49,60,45),(16,68,62,25),(17,52,63,39),(21,47,64,43),(24,50,67,37),(27,53,70,40)], [(1,47,30,43),(2,22,31,65),(3,60,32,14),(4,50,33,37),(5,25,34,68),(6,63,35,17),(7,53,36,40),(8,19,28,71),(9,57,29,11),(10,54,56,41),(12,64,58,21),(13,48,59,44),(15,67,61,24),(16,51,62,38),(18,70,55,27),(20,46,72,42),(23,49,66,45),(26,52,69,39)], [(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 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F9A···9F12A···12H18A···18F
order12223344446666669···912···1218···18
size113311666611333316···166···616···16

36 irreducible representations

dim111222333366
type+-++-
imageC1C3C9SL2(𝔽3)SL2(𝔽3)Q8⋊C9A4A4C3.A4C3.A4Q8⋊A4C22⋊(Q8⋊C9)
kernelC22⋊(Q8⋊C9)Q8×C2×C6C22×Q8C2×C6C2×C6C22C3×Q8C22×C6Q8C23C3C1
# reps126126418212

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

10000
01000
003600
003601
003610
,
10000
01000
000361
000360
001360
,
2726000
2610000
00100
00010
00001
,
01000
360000
00100
00010
00001
,
10000
2610000
00009
00900
00090

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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