metabelian, supersoluble, monomial, A-group
Aliases: C11⋊D11, C112⋊2C2, SmallGroup(242,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C112 — C11⋊D11 |
Generators and relations for C11⋊D11
G = < a,b,c | a11=b11=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C11⋊D11
►On 121 points
Generators in S121
(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)(111 112 113 114 115 116 117 118 119 120 121)
(1 41 94 108 48 67 18 116 56 86 26)(2 42 95 109 49 68 19 117 57 87 27)(3 43 96 110 50 69 20 118 58 88 28)(4 44 97 100 51 70 21 119 59 78 29)(5 34 98 101 52 71 22 120 60 79 30)(6 35 99 102 53 72 12 121 61 80 31)(7 36 89 103 54 73 13 111 62 81 32)(8 37 90 104 55 74 14 112 63 82 33)(9 38 91 105 45 75 15 113 64 83 23)(10 39 92 106 46 76 16 114 65 84 24)(11 40 93 107 47 77 17 115 66 85 25)
(1 26)(2 25)(3 24)(4 23)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 55)(34 82)(35 81)(36 80)(37 79)(38 78)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 99)(63 98)(64 97)(65 96)(66 95)(68 77)(69 76)(70 75)(71 74)(72 73)(100 113)(101 112)(102 111)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)
G:=sub<Sym(121)| (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)(111,112,113,114,115,116,117,118,119,120,121), (1,41,94,108,48,67,18,116,56,86,26)(2,42,95,109,49,68,19,117,57,87,27)(3,43,96,110,50,69,20,118,58,88,28)(4,44,97,100,51,70,21,119,59,78,29)(5,34,98,101,52,71,22,120,60,79,30)(6,35,99,102,53,72,12,121,61,80,31)(7,36,89,103,54,73,13,111,62,81,32)(8,37,90,104,55,74,14,112,63,82,33)(9,38,91,105,45,75,15,113,64,83,23)(10,39,92,106,46,76,16,114,65,84,24)(11,40,93,107,47,77,17,115,66,85,25), (1,26)(2,25)(3,24)(4,23)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,55)(34,82)(35,81)(36,80)(37,79)(38,78)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,99)(63,98)(64,97)(65,96)(66,95)(68,77)(69,76)(70,75)(71,74)(72,73)(100,113)(101,112)(102,111)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)>;
G:=Group( (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)(111,112,113,114,115,116,117,118,119,120,121), (1,41,94,108,48,67,18,116,56,86,26)(2,42,95,109,49,68,19,117,57,87,27)(3,43,96,110,50,69,20,118,58,88,28)(4,44,97,100,51,70,21,119,59,78,29)(5,34,98,101,52,71,22,120,60,79,30)(6,35,99,102,53,72,12,121,61,80,31)(7,36,89,103,54,73,13,111,62,81,32)(8,37,90,104,55,74,14,112,63,82,33)(9,38,91,105,45,75,15,113,64,83,23)(10,39,92,106,46,76,16,114,65,84,24)(11,40,93,107,47,77,17,115,66,85,25), (1,26)(2,25)(3,24)(4,23)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,55)(34,82)(35,81)(36,80)(37,79)(38,78)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,99)(63,98)(64,97)(65,96)(66,95)(68,77)(69,76)(70,75)(71,74)(72,73)(100,113)(101,112)(102,111)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114) );
G=PermutationGroup([[(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),(111,112,113,114,115,116,117,118,119,120,121)], [(1,41,94,108,48,67,18,116,56,86,26),(2,42,95,109,49,68,19,117,57,87,27),(3,43,96,110,50,69,20,118,58,88,28),(4,44,97,100,51,70,21,119,59,78,29),(5,34,98,101,52,71,22,120,60,79,30),(6,35,99,102,53,72,12,121,61,80,31),(7,36,89,103,54,73,13,111,62,81,32),(8,37,90,104,55,74,14,112,63,82,33),(9,38,91,105,45,75,15,113,64,83,23),(10,39,92,106,46,76,16,114,65,84,24),(11,40,93,107,47,77,17,115,66,85,25)], [(1,26),(2,25),(3,24),(4,23),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,55),(34,82),(35,81),(36,80),(37,79),(38,78),(39,88),(40,87),(41,86),(42,85),(43,84),(44,83),(56,94),(57,93),(58,92),(59,91),(60,90),(61,89),(62,99),(63,98),(64,97),(65,96),(66,95),(68,77),(69,76),(70,75),(71,74),(72,73),(100,113),(101,112),(102,111),(103,121),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114)]])
C11⋊D11 is a maximal subgroup of
C112⋊C4 D112
C11⋊D11 is a maximal quotient of C11⋊Dic11
62 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11BH |
| order | 1 | 2 | 11 | ··· | 11 |
| size | 1 | 121 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D11 |
| kernel | C11⋊D11 | C112 | C11 |
| # reps | 1 | 1 | 60 |
Matrix representation of C11⋊D11 ►in GL4(𝔽23) generated by
| 2 | 12 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 14 |
| 0 | 0 | 20 | 7 |
| 0 | 1 | 0 | 0 |
| 22 | 14 | 0 | 0 |
| 0 | 0 | 9 | 1 |
| 0 | 0 | 8 | 1 |
| 12 | 14 | 0 | 0 |
| 21 | 11 | 0 | 0 |
| 0 | 0 | 1 | 22 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(23))| [2,11,0,0,12,9,0,0,0,0,4,20,0,0,14,7],[0,22,0,0,1,14,0,0,0,0,9,8,0,0,1,1],[12,21,0,0,14,11,0,0,0,0,1,0,0,0,22,22] >;
C11⋊D11 in GAP, Magma, Sage, TeX
C_{11}\rtimes D_{11} % in TeX
G:=Group("C11:D11"); // GroupNames label
G:=SmallGroup(242,4);
// by ID
G=gap.SmallGroup(242,4);
# by ID
G:=PCGroup([3,-2,-11,-11,121,1982]);
// Polycyclic
G:=Group<a,b,c|a^11=b^11=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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