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## G = C122⋊16C2order 288 = 25·32

### 16th semidirect product of C122 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C122⋊16C2
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C122⋊16C2
 Lower central C32 — C3×C6 — C122⋊16C2
 Upper central C1 — C2×C4 — C42

Generators and relations for C12216C2
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5, cbc=a6b5 >

Subgroups: 772 in 228 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C32, Dic3 [×16], C12 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×16], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C42⋊C2, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×4], Dic3⋊C4 [×8], D6⋊C4 [×8], C4×C12 [×4], S3×C2×C4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, C422S3 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C6.11D12 [×2], C122, C2×C4×C3⋊S3, C12216C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C4○D4 [×2], C3⋊S3, C4×S3 [×8], C22×S3 [×4], C42⋊C2, C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4○D12 [×8], C4×C3⋊S3 [×2], C22×C3⋊S3, C422S3 [×4], C2×C4×C3⋊S3, C12.59D6 [×2], C12216C2

Smallest permutation representation of C12216C2
On 144 points
Generators in S144
(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)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 136 17 76 126 55 45 65 94 102 25 114)(2 137 18 77 127 56 46 66 95 103 26 115)(3 138 19 78 128 57 47 67 96 104 27 116)(4 139 20 79 129 58 48 68 85 105 28 117)(5 140 21 80 130 59 37 69 86 106 29 118)(6 141 22 81 131 60 38 70 87 107 30 119)(7 142 23 82 132 49 39 71 88 108 31 120)(8 143 24 83 121 50 40 72 89 97 32 109)(9 144 13 84 122 51 41 61 90 98 33 110)(10 133 14 73 123 52 42 62 91 99 34 111)(11 134 15 74 124 53 43 63 92 100 35 112)(12 135 16 75 125 54 44 64 93 101 36 113)
(2 6)(3 11)(5 9)(8 12)(13 29)(14 34)(15 27)(16 32)(17 25)(18 30)(19 35)(20 28)(21 33)(22 26)(23 31)(24 36)(37 41)(38 46)(40 44)(43 47)(49 136)(50 141)(51 134)(52 139)(53 144)(54 137)(55 142)(56 135)(57 140)(58 133)(59 138)(60 143)(61 112)(62 117)(63 110)(64 115)(65 120)(66 113)(67 118)(68 111)(69 116)(70 109)(71 114)(72 119)(73 79)(74 84)(75 77)(76 82)(78 80)(81 83)(85 129)(86 122)(87 127)(88 132)(89 125)(90 130)(91 123)(92 128)(93 121)(94 126)(95 131)(96 124)(97 107)(98 100)(99 105)(101 103)(102 108)(104 106)

G:=sub<Sym(144)| (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,136,17,76,126,55,45,65,94,102,25,114)(2,137,18,77,127,56,46,66,95,103,26,115)(3,138,19,78,128,57,47,67,96,104,27,116)(4,139,20,79,129,58,48,68,85,105,28,117)(5,140,21,80,130,59,37,69,86,106,29,118)(6,141,22,81,131,60,38,70,87,107,30,119)(7,142,23,82,132,49,39,71,88,108,31,120)(8,143,24,83,121,50,40,72,89,97,32,109)(9,144,13,84,122,51,41,61,90,98,33,110)(10,133,14,73,123,52,42,62,91,99,34,111)(11,134,15,74,124,53,43,63,92,100,35,112)(12,135,16,75,125,54,44,64,93,101,36,113), (2,6)(3,11)(5,9)(8,12)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(37,41)(38,46)(40,44)(43,47)(49,136)(50,141)(51,134)(52,139)(53,144)(54,137)(55,142)(56,135)(57,140)(58,133)(59,138)(60,143)(61,112)(62,117)(63,110)(64,115)(65,120)(66,113)(67,118)(68,111)(69,116)(70,109)(71,114)(72,119)(73,79)(74,84)(75,77)(76,82)(78,80)(81,83)(85,129)(86,122)(87,127)(88,132)(89,125)(90,130)(91,123)(92,128)(93,121)(94,126)(95,131)(96,124)(97,107)(98,100)(99,105)(101,103)(102,108)(104,106)>;

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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,136,17,76,126,55,45,65,94,102,25,114)(2,137,18,77,127,56,46,66,95,103,26,115)(3,138,19,78,128,57,47,67,96,104,27,116)(4,139,20,79,129,58,48,68,85,105,28,117)(5,140,21,80,130,59,37,69,86,106,29,118)(6,141,22,81,131,60,38,70,87,107,30,119)(7,142,23,82,132,49,39,71,88,108,31,120)(8,143,24,83,121,50,40,72,89,97,32,109)(9,144,13,84,122,51,41,61,90,98,33,110)(10,133,14,73,123,52,42,62,91,99,34,111)(11,134,15,74,124,53,43,63,92,100,35,112)(12,135,16,75,125,54,44,64,93,101,36,113), (2,6)(3,11)(5,9)(8,12)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(37,41)(38,46)(40,44)(43,47)(49,136)(50,141)(51,134)(52,139)(53,144)(54,137)(55,142)(56,135)(57,140)(58,133)(59,138)(60,143)(61,112)(62,117)(63,110)(64,115)(65,120)(66,113)(67,118)(68,111)(69,116)(70,109)(71,114)(72,119)(73,79)(74,84)(75,77)(76,82)(78,80)(81,83)(85,129)(86,122)(87,127)(88,132)(89,125)(90,130)(91,123)(92,128)(93,121)(94,126)(95,131)(96,124)(97,107)(98,100)(99,105)(101,103)(102,108)(104,106) );

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),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,136,17,76,126,55,45,65,94,102,25,114),(2,137,18,77,127,56,46,66,95,103,26,115),(3,138,19,78,128,57,47,67,96,104,27,116),(4,139,20,79,129,58,48,68,85,105,28,117),(5,140,21,80,130,59,37,69,86,106,29,118),(6,141,22,81,131,60,38,70,87,107,30,119),(7,142,23,82,132,49,39,71,88,108,31,120),(8,143,24,83,121,50,40,72,89,97,32,109),(9,144,13,84,122,51,41,61,90,98,33,110),(10,133,14,73,123,52,42,62,91,99,34,111),(11,134,15,74,124,53,43,63,92,100,35,112),(12,135,16,75,125,54,44,64,93,101,36,113)], [(2,6),(3,11),(5,9),(8,12),(13,29),(14,34),(15,27),(16,32),(17,25),(18,30),(19,35),(20,28),(21,33),(22,26),(23,31),(24,36),(37,41),(38,46),(40,44),(43,47),(49,136),(50,141),(51,134),(52,139),(53,144),(54,137),(55,142),(56,135),(57,140),(58,133),(59,138),(60,143),(61,112),(62,117),(63,110),(64,115),(65,120),(66,113),(67,118),(68,111),(69,116),(70,109),(71,114),(72,119),(73,79),(74,84),(75,77),(76,82),(78,80),(81,83),(85,129),(86,122),(87,127),(88,132),(89,125),(90,130),(91,123),(92,128),(93,121),(94,126),(95,131),(96,124),(97,107),(98,100),(99,105),(101,103),(102,108),(104,106)])

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A ··· 6L 12A ··· 12AV order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 2 2 2 2 1 1 1 1 2 2 2 2 18 ··· 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D6 C4○D4 C4×S3 C4○D12 kernel C122⋊16C2 C4×C3⋊Dic3 C6.Dic6 C6.11D12 C122 C2×C4×C3⋊S3 C4×C3⋊S3 C4×C12 C2×C12 C3×C6 C12 C6 # reps 1 1 2 2 1 1 8 4 12 4 16 32

Matrix representation of C12216C2 in GL5(𝔽13)

 12 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8 5 0 0 0 8 0
,
 5 0 0 0 0 0 2 2 0 0 0 11 4 0 0 0 0 0 11 4 0 0 0 9 2
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,8,0,0,0,5,0],[5,0,0,0,0,0,2,11,0,0,0,2,4,0,0,0,0,0,11,9,0,0,0,4,2],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C12216C2 in GAP, Magma, Sage, TeX

C_{12}^2\rtimes_{16}C_2
% in TeX

G:=Group("C12^2:16C2");
// GroupNames label

G:=SmallGroup(288,729);
// by ID

G=gap.SmallGroup(288,729);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=a^6*b^5>;
// generators/relations

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