direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C2.F9, C62.1C8, C22.2F9, (C3×C6)⋊C16, C2.3(C2×F9), C32⋊2(C2×C16), C3⋊Dic3.1C8, C32⋊2C8.2C4, C32⋊2C8.5C22, (C3×C6).5(C2×C8), (C2×C3⋊Dic3).4C4, C3⋊Dic3.3(C2×C4), (C2×C32⋊2C8).4C2, SmallGroup(288,865)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C2.F9 |
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 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 81)(15 82)(16 83)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 80)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)
(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 39 55)(3 40 56)(4 57 41)(6 59 43)(7 60 44)(8 45 61)(10 47 63)(11 48 64)(12 49 33)(14 51 35)(15 52 36)(16 37 53)(17 70 85)(18 71 86)(19 87 72)(21 89 74)(22 90 75)(23 76 91)(25 78 93)(26 79 94)(27 95 80)(29 81 66)(30 82 67)(31 68 83)
(1 38 54)(3 40 56)(4 41 57)(5 58 42)(7 60 44)(8 61 45)(9 46 62)(11 48 64)(12 33 49)(13 50 34)(15 52 36)(16 53 37)(18 71 86)(19 72 87)(20 88 73)(22 90 75)(23 91 76)(24 77 92)(26 79 94)(27 80 95)(28 96 65)(30 82 67)(31 83 68)(32 69 84)
(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)| (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,81)(15,82)(16,83)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,80)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79), (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), (2,39,55)(3,40,56)(4,57,41)(6,59,43)(7,60,44)(8,45,61)(10,47,63)(11,48,64)(12,49,33)(14,51,35)(15,52,36)(16,37,53)(17,70,85)(18,71,86)(19,87,72)(21,89,74)(22,90,75)(23,76,91)(25,78,93)(26,79,94)(27,95,80)(29,81,66)(30,82,67)(31,68,83), (1,38,54)(3,40,56)(4,41,57)(5,58,42)(7,60,44)(8,61,45)(9,46,62)(11,48,64)(12,33,49)(13,50,34)(15,52,36)(16,53,37)(18,71,86)(19,72,87)(20,88,73)(22,90,75)(23,91,76)(24,77,92)(26,79,94)(27,80,95)(28,96,65)(30,82,67)(31,83,68)(32,69,84), (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( (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,81)(15,82)(16,83)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,80)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79), (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), (2,39,55)(3,40,56)(4,57,41)(6,59,43)(7,60,44)(8,45,61)(10,47,63)(11,48,64)(12,49,33)(14,51,35)(15,52,36)(16,37,53)(17,70,85)(18,71,86)(19,87,72)(21,89,74)(22,90,75)(23,76,91)(25,78,93)(26,79,94)(27,95,80)(29,81,66)(30,82,67)(31,68,83), (1,38,54)(3,40,56)(4,41,57)(5,58,42)(7,60,44)(8,61,45)(9,46,62)(11,48,64)(12,33,49)(13,50,34)(15,52,36)(16,53,37)(18,71,86)(19,72,87)(20,88,73)(22,90,75)(23,91,76)(24,77,92)(26,79,94)(27,80,95)(28,96,65)(30,82,67)(31,83,68)(32,69,84), (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([[(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,81),(15,82),(16,83),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,80),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79)], [(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)], [(2,39,55),(3,40,56),(4,57,41),(6,59,43),(7,60,44),(8,45,61),(10,47,63),(11,48,64),(12,49,33),(14,51,35),(15,52,36),(16,37,53),(17,70,85),(18,71,86),(19,87,72),(21,89,74),(22,90,75),(23,76,91),(25,78,93),(26,79,94),(27,95,80),(29,81,66),(30,82,67),(31,68,83)], [(1,38,54),(3,40,56),(4,41,57),(5,58,42),(7,60,44),(8,61,45),(9,46,62),(11,48,64),(12,33,49),(13,50,34),(15,52,36),(16,53,37),(18,71,86),(19,72,87),(20,88,73),(22,90,75),(23,91,76),(24,77,92),(26,79,94),(27,80,95),(28,96,65),(30,82,67),(31,83,68),(32,69,84)], [(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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