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G = C2×C2.F9order 288 = 25·32

Direct product of C2 and C2.F9

Aliases: C2×C2.F9, C62.1C8, C22.2F9, (C3×C6)⋊C16, C2.3(C2×F9), C322(C2×C16), C3⋊Dic3.1C8, C322C8.2C4, C322C8.5C22, (C3×C6).5(C2×C8), (C2×C3⋊Dic3).4C4, C3⋊Dic3.3(C2×C4), (C2×C322C8).4C2, SmallGroup(288,865)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C2.F9
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2.F9 — C2×C2.F9
 Lower central C32 — C2×C2.F9
 Upper central C1 — C22

Generators and relations for C2×C2.F9
G = < a,b,c,d,e | a2=b2=c3=d3=1, e8=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C2×C2.F9
On 96 points
Generators in S96
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 33)(14 34)(15 35)(16 36)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 65)(31 66)(32 67)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 81)(64 82)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)
(2 73 62)(3 74 63)(4 64 75)(6 50 77)(7 51 78)(8 79 52)(10 65 54)(11 66 55)(12 56 67)(14 58 69)(15 59 70)(16 71 60)(18 34 92)(19 35 93)(20 94 36)(22 96 38)(23 81 39)(24 40 82)(26 42 84)(27 43 85)(28 86 44)(30 88 46)(31 89 47)(32 48 90)
(1 72 61)(3 74 63)(4 75 64)(5 49 76)(7 51 78)(8 52 79)(9 80 53)(11 66 55)(12 67 56)(13 57 68)(15 59 70)(16 60 71)(17 33 91)(19 35 93)(20 36 94)(21 95 37)(23 81 39)(24 82 40)(25 41 83)(27 43 85)(28 44 86)(29 87 45)(31 89 47)(32 90 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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,33)(14,34)(15,35)(16,36)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,65)(31,66)(32,67)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,81)(64,82), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (2,73,62)(3,74,63)(4,64,75)(6,50,77)(7,51,78)(8,79,52)(10,65,54)(11,66,55)(12,56,67)(14,58,69)(15,59,70)(16,71,60)(18,34,92)(19,35,93)(20,94,36)(22,96,38)(23,81,39)(24,40,82)(26,42,84)(27,43,85)(28,86,44)(30,88,46)(31,89,47)(32,48,90), (1,72,61)(3,74,63)(4,75,64)(5,49,76)(7,51,78)(8,52,79)(9,80,53)(11,66,55)(12,67,56)(13,57,68)(15,59,70)(16,60,71)(17,33,91)(19,35,93)(20,36,94)(21,95,37)(23,81,39)(24,82,40)(25,41,83)(27,43,85)(28,44,86)(29,87,45)(31,89,47)(32,90,48), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,33)(14,34)(15,35)(16,36)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,65)(31,66)(32,67)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,81)(64,82), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (2,73,62)(3,74,63)(4,64,75)(6,50,77)(7,51,78)(8,79,52)(10,65,54)(11,66,55)(12,56,67)(14,58,69)(15,59,70)(16,71,60)(18,34,92)(19,35,93)(20,94,36)(22,96,38)(23,81,39)(24,40,82)(26,42,84)(27,43,85)(28,86,44)(30,88,46)(31,89,47)(32,48,90), (1,72,61)(3,74,63)(4,75,64)(5,49,76)(7,51,78)(8,52,79)(9,80,53)(11,66,55)(12,67,56)(13,57,68)(15,59,70)(16,60,71)(17,33,91)(19,35,93)(20,36,94)(21,95,37)(23,81,39)(24,82,40)(25,41,83)(27,43,85)(28,44,86)(29,87,45)(31,89,47)(32,90,48), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,33),(14,34),(15,35),(16,36),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,65),(31,66),(32,67),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,95),(62,96),(63,81),(64,82)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96)], [(2,73,62),(3,74,63),(4,64,75),(6,50,77),(7,51,78),(8,79,52),(10,65,54),(11,66,55),(12,56,67),(14,58,69),(15,59,70),(16,71,60),(18,34,92),(19,35,93),(20,94,36),(22,96,38),(23,81,39),(24,40,82),(26,42,84),(27,43,85),(28,86,44),(30,88,46),(31,89,47),(32,48,90)], [(1,72,61),(3,74,63),(4,75,64),(5,49,76),(7,51,78),(8,52,79),(9,80,53),(11,66,55),(12,67,56),(13,57,68),(15,59,70),(16,60,71),(17,33,91),(19,35,93),(20,36,94),(21,95,37),(23,81,39),(24,82,40),(25,41,83),(27,43,85),(28,44,86),(29,87,45),(31,89,47),(32,90,48)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)])

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 8A ··· 8H 16A ··· 16P order 1 2 2 2 3 4 4 4 4 6 6 6 8 ··· 8 16 ··· 16 size 1 1 1 1 8 9 9 9 9 8 8 8 9 ··· 9 9 ··· 9

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 8 8 8 type + + + + - + image C1 C2 C2 C4 C4 C8 C8 C16 F9 C2.F9 C2×F9 kernel C2×C2.F9 C2.F9 C2×C32⋊2C8 C32⋊2C8 C2×C3⋊Dic3 C3⋊Dic3 C62 C3×C6 C22 C2 C2 # reps 1 2 1 2 2 4 4 16 1 2 1

Matrix representation of C2×C2.F9 in GL10(𝔽97)

 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 96 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 96 0 0 0 0 0 81 58 68 0 0 47 96 96 0 0 0 0 0 29 50 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 1 96 0 0 0 0 56 40 0 29 0 50 1 0 0 0 56 40 0 29 0 50 0 1
,
 85 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 96 1 0 0 81 58 68 68 47 47 95 96 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 67 5 5 90 50 29 0 0 0 71 82 5 5 90 50 29 0

G:=sub<GL(10,GF(97))| [96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,81,0,0,0,0,1,0,0,0,0,58,0,0,0,0,0,0,1,0,0,68,0,0,0,0,0,96,96,0,0,0,29,0,0,0,0,0,0,96,96,0,50,0,0,0,0,0,0,1,0,47,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,0,96,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,96,96,0,0,0,0,56,56,0,0,1,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,0,96,96,0,0,29,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,96,96,50,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[85,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,0,0,81,0,0,16,71,0,0,0,0,0,58,0,0,67,82,0,0,0,0,0,68,1,0,5,5,0,0,0,0,0,68,0,1,5,5,0,0,1,0,0,47,0,0,90,90,0,0,0,1,0,47,0,0,50,50,0,0,0,0,96,95,0,0,29,29,0,0,0,0,1,96,0,0,0,0] >;

C2×C2.F9 in GAP, Magma, Sage, TeX

C_2\times C_2.F_9
% in TeX

G:=Group("C2xC2.F9");
// GroupNames label

G:=SmallGroup(288,865);
// by ID

G=gap.SmallGroup(288,865);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=1,e^8=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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