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G = C2xD5xD11order 440 = 23·5·11

Direct product of C2, D5 and D11

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2xD5xD11, C55:C23, C10:1D22, C22:1D10, C110:C22, D55:C22, D110:5C2, (D5xC22):3C2, (D5xC11):C22, (C5xD11):C22, C5:1(C22xD11), C11:1(C22xD5), (C10xD11):3C2, SmallGroup(440,47)

Series: Derived Chief Lower central Upper central

C1C55 — C2xD5xD11
C1C11C55C5xD11D5xD11 — C2xD5xD11
C55 — C2xD5xD11
C1C2

Generators and relations for C2xD5xD11
 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, C2xC10, D11, D11, C22, C22, C22xD5, D22, D22, C2xC22, C55, C22xD11, D5xC11, C5xD11, D55, C110, D5xD11, C10xD11, D5xC22, D110, C2xD5xD11
Quotients: C1, C2, C22, C23, D5, D10, D11, C22xD5, D22, C22xD11, D5xD11, C2xD5xD11

Smallest permutation representation of C2xD5xD11
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 2A2B2C2D2E2F2G5A5B10A10B10C10D10E10F11A···11E22A···22E22F···22O55A···55J110A···110J
order122222225510101010101011···1122···2222···2255···55110···110
size1155111155552222222222222···22···210···104···44···4

56 irreducible representations

dim1111122222244
type+++++++++++++
imageC1C2C2C2C2D5D10D10D11D22D22D5xD11C2xD5xD11
kernelC2xD5xD11D5xD11C10xD11D5xC22D110D22D11C22D10D5C10C2C1
# reps1411124251051010

Matrix representation of C2xD5xD11 in GL4(F331) generated by

330000
033000
0010
0001
,
1000
0100
00115210
0017499
,
330000
033000
003300
001641
,
323100
32120800
0010
0001
,
18622400
4814500
0010
0001
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] >;

C2xD5xD11 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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