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## G = C11⋊D11order 242 = 2·112

### The semidirect product of C11 and D11 acting via D11/C11=C2

Aliases: C11⋊D11, C1122C2, SmallGroup(242,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112 — C11⋊D11
 Chief series C1 — C11 — C112 — C11⋊D11
 Lower central C112 — C11⋊D11
 Upper central C1

Generators and relations for C11⋊D11
G = < a,b,c | a11=b11=c2=1, ab=ba, cac=a-1, cbc=b-1 >

121C2
11D11
11D11
11D11
11D11
11D11
11D11
11D11
11D11
11D11
11D11
11D11
11D11

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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