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## G = C2×C11⋊F5order 440 = 23·5·11

### Direct product of C2 and C11⋊F5

Aliases: C2×C11⋊F5, C22⋊F5, C1101C4, D5⋊Dic11, D10.D11, C10⋊Dic11, D5.2D22, C552(C2×C4), C5⋊(C2×Dic11), C112(C2×F5), (D5×C11)⋊2C4, (D5×C22).2C2, (D5×C11).2C22, SmallGroup(440,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — C2×C11⋊F5
 Chief series C1 — C11 — C55 — D5×C11 — C11⋊F5 — C2×C11⋊F5
 Lower central C55 — C2×C11⋊F5
 Upper central C1 — C2

Generators and relations for C2×C11⋊F5
G = < a,b,c,d | a2=b11=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C2×C11⋊F5
On 110 points
Generators in S110
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)
(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 97 98 99)(100 101 102 103 104 105 106 107 108 109 110)
(1 45 34 23 12)(2 46 35 24 13)(3 47 36 25 14)(4 48 37 26 15)(5 49 38 27 16)(6 50 39 28 17)(7 51 40 29 18)(8 52 41 30 19)(9 53 42 31 20)(10 54 43 32 21)(11 55 44 33 22)(56 100 89 78 67)(57 101 90 79 68)(58 102 91 80 69)(59 103 92 81 70)(60 104 93 82 71)(61 105 94 83 72)(62 106 95 84 73)(63 107 96 85 74)(64 108 97 86 75)(65 109 98 87 76)(66 110 99 88 77)
(1 56)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 78 45 89)(13 88 46 99)(14 87 47 98)(15 86 48 97)(16 85 49 96)(17 84 50 95)(18 83 51 94)(19 82 52 93)(20 81 53 92)(21 80 54 91)(22 79 55 90)(23 100 34 67)(24 110 35 77)(25 109 36 76)(26 108 37 75)(27 107 38 74)(28 106 39 73)(29 105 40 72)(30 104 41 71)(31 103 42 70)(32 102 43 69)(33 101 44 68)

G:=sub<Sym(110)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110), (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)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110), (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22)(56,100,89,78,67)(57,101,90,79,68)(58,102,91,80,69)(59,103,92,81,70)(60,104,93,82,71)(61,105,94,83,72)(62,106,95,84,73)(63,107,96,85,74)(64,108,97,86,75)(65,109,98,87,76)(66,110,99,88,77), (1,56)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,78,45,89)(13,88,46,99)(14,87,47,98)(15,86,48,97)(16,85,49,96)(17,84,50,95)(18,83,51,94)(19,82,52,93)(20,81,53,92)(21,80,54,91)(22,79,55,90)(23,100,34,67)(24,110,35,77)(25,109,36,76)(26,108,37,75)(27,107,38,74)(28,106,39,73)(29,105,40,72)(30,104,41,71)(31,103,42,70)(32,102,43,69)(33,101,44,68)>;

G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110), (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)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110), (1,45,34,23,12)(2,46,35,24,13)(3,47,36,25,14)(4,48,37,26,15)(5,49,38,27,16)(6,50,39,28,17)(7,51,40,29,18)(8,52,41,30,19)(9,53,42,31,20)(10,54,43,32,21)(11,55,44,33,22)(56,100,89,78,67)(57,101,90,79,68)(58,102,91,80,69)(59,103,92,81,70)(60,104,93,82,71)(61,105,94,83,72)(62,106,95,84,73)(63,107,96,85,74)(64,108,97,86,75)(65,109,98,87,76)(66,110,99,88,77), (1,56)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,78,45,89)(13,88,46,99)(14,87,47,98)(15,86,48,97)(16,85,49,96)(17,84,50,95)(18,83,51,94)(19,82,52,93)(20,81,53,92)(21,80,54,91)(22,79,55,90)(23,100,34,67)(24,110,35,77)(25,109,36,76)(26,108,37,75)(27,107,38,74)(28,106,39,73)(29,105,40,72)(30,104,41,71)(31,103,42,70)(32,102,43,69)(33,101,44,68) );

G=PermutationGroup([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110)], [(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),(89,90,91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108,109,110)], [(1,45,34,23,12),(2,46,35,24,13),(3,47,36,25,14),(4,48,37,26,15),(5,49,38,27,16),(6,50,39,28,17),(7,51,40,29,18),(8,52,41,30,19),(9,53,42,31,20),(10,54,43,32,21),(11,55,44,33,22),(56,100,89,78,67),(57,101,90,79,68),(58,102,91,80,69),(59,103,92,81,70),(60,104,93,82,71),(61,105,94,83,72),(62,106,95,84,73),(63,107,96,85,74),(64,108,97,86,75),(65,109,98,87,76),(66,110,99,88,77)], [(1,56),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,78,45,89),(13,88,46,99),(14,87,47,98),(15,86,48,97),(16,85,49,96),(17,84,50,95),(18,83,51,94),(19,82,52,93),(20,81,53,92),(21,80,54,91),(22,79,55,90),(23,100,34,67),(24,110,35,77),(25,109,36,76),(26,108,37,75),(27,107,38,74),(28,106,39,73),(29,105,40,72),(30,104,41,71),(31,103,42,70),(32,102,43,69),(33,101,44,68)]])

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5 10 11A ··· 11E 22A ··· 22E 22F ··· 22O 55A ··· 55J 110A ··· 110J order 1 2 2 2 4 4 4 4 5 10 11 ··· 11 22 ··· 22 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 5 5 55 55 55 55 4 4 2 ··· 2 2 ··· 2 10 ··· 10 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 D11 Dic11 D22 Dic11 F5 C2×F5 C11⋊F5 C2×C11⋊F5 kernel C2×C11⋊F5 C11⋊F5 D5×C22 D5×C11 C110 D10 D5 D5 C10 C22 C11 C2 C1 # reps 1 2 1 2 2 5 5 5 5 1 1 10 10

Matrix representation of C2×C11⋊F5 in GL6(𝔽661)

 660 0 0 0 0 0 0 660 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 660 0 0 0 0 556 520 0 0 0 0 0 0 217 660 0 0 0 0 1 0 0 0 0 0 0 0 217 660 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 660 0 506 370 0 0 0 660 291 154
,
 422 49 0 0 0 0 480 239 0 0 0 0 0 0 1 0 0 0 0 0 217 660 0 0 0 0 155 291 155 291 0 0 215 506 215 506

G:=sub<GL(6,GF(661))| [660,0,0,0,0,0,0,660,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,556,0,0,0,0,660,520,0,0,0,0,0,0,217,1,0,0,0,0,660,0,0,0,0,0,0,0,217,1,0,0,0,0,660,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,1,0,506,291,0,0,0,1,370,154],[422,480,0,0,0,0,49,239,0,0,0,0,0,0,1,217,155,215,0,0,0,660,291,506,0,0,0,0,155,215,0,0,0,0,291,506] >;

C2×C11⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_{11}\rtimes F_5
% in TeX

G:=Group("C2xC11:F5");
// GroupNames label

G:=SmallGroup(440,46);
// by ID

G=gap.SmallGroup(440,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,20,483,173,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^11=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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