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## G = C2×D5×D11order 440 = 23·5·11

### Direct product of C2, D5 and D11

Aliases: C2×D5×D11, C55⋊C23, C101D22, C221D10, C110⋊C22, D55⋊C22, D1105C2, (D5×C22)⋊3C2, (D5×C11)⋊C22, (C5×D11)⋊C22, C51(C22×D11), C111(C22×D5), (C10×D11)⋊3C2, SmallGroup(440,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — C2×D5×D11
 Chief series C1 — C11 — C55 — C5×D11 — D5×D11 — C2×D5×D11
 Lower central C55 — C2×D5×D11
 Upper central C1 — C2

Generators and relations for C2×D5×D11
G = < a,b,c,d,e | a2=b5=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 692 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C22, C5, C23, D5, D5, C10, C10, C11, D10, D10, C2×C10, D11, D11, C22, C22, C22×D5, D22, D22, C2×C22, C55, C22×D11, D5×C11, C5×D11, D55, C110, D5×D11, C10×D11, D5×C22, D110, C2×D5×D11
Quotients: C1, C2, C22, C23, D5, D10, D11, C22×D5, D22, C22×D11, D5×D11, C2×D5×D11

Smallest permutation representation of C2×D5×D11
On 110 points
Generators in S110
(1 65)(2 66)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(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 54 43 32 21)(2 55 44 33 22)(3 45 34 23 12)(4 46 35 24 13)(5 47 36 25 14)(6 48 37 26 15)(7 49 38 27 16)(8 50 39 28 17)(9 51 40 29 18)(10 52 41 30 19)(11 53 42 31 20)(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 76)(2 77)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)(100 106)(101 105)(102 104)(107 110)(108 109)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D5 D10 D10 D11 D22 D22 D5×D11 C2×D5×D11 kernel C2×D5×D11 D5×D11 C10×D11 D5×C22 D110 D22 D11 C22 D10 D5 C10 C2 C1 # reps 1 4 1 1 1 2 4 2 5 10 5 10 10

Matrix representation of C2×D5×D11 in GL4(𝔽331) generated by

 330 0 0 0 0 330 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 115 210 0 0 174 99
,
 330 0 0 0 0 330 0 0 0 0 330 0 0 0 164 1
,
 323 1 0 0 321 208 0 0 0 0 1 0 0 0 0 1
,
 186 224 0 0 48 145 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(331))| [330,0,0,0,0,330,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,115,174,0,0,210,99],[330,0,0,0,0,330,0,0,0,0,330,164,0,0,0,1],[323,321,0,0,1,208,0,0,0,0,1,0,0,0,0,1],[186,48,0,0,224,145,0,0,0,0,1,0,0,0,0,1] >;

C2×D5×D11 in GAP, Magma, Sage, TeX

C_2\times D_5\times D_{11}
% in TeX

G:=Group("C2xD5xD11");
// GroupNames label

G:=SmallGroup(440,47);
// by ID

G=gap.SmallGroup(440,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,328,10004]);
// Polycyclic

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

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