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G = C2×C2.F9order 288 = 25·32

Direct product of C2 and C2.F9

direct product, metabelian, soluble, monomial, A-group

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

C1C32 — C2×C2.F9
C1C32C3×C6C3⋊Dic3C322C8C2.F9 — C2×C2.F9
C32 — C2×C2.F9
C1C22

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 >

4C3
9C4
9C4
4C6
4C6
4C6
9C8
9C2×C4
9C8
4C2×C6
12Dic3
12Dic3
9C16
9C16
9C2×C8
12C2×Dic3
9C2×C16

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 2A2B2C 3 4A4B4C4D6A6B6C8A···8H16A···16P
order1222344446668···816···16
size1111899998889···99···9

36 irreducible representations

dim11111111888
type++++-+
imageC1C2C2C4C4C8C8C16F9C2.F9C2×F9
kernelC2×C2.F9C2.F9C2×C322C8C322C8C2×C3⋊Dic3C3⋊Dic3C62C3×C6C22C2C2
# reps121224416121

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

96000000000
09600000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
96000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
00000960000
00001960000
00000096100
00000096000
0081586800479696
000002950010
,
1000000000
0100000000
00961000000
00960000000
00000960000
00001960000
00000009600
00000019600
00564002905010
00564002905001
,
85000000000
02200000000
0000001000
0000000100
00000000961
008158686847479596
0000100000
0000010000
001667559050290
007182559050290

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

Export

Subgroup lattice of C2×C2.F9 in TeX

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