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## G = C2×C32⋊11SD16order 288 = 25·32

### Direct product of C2 and C32⋊11SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊11SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C2×C32⋊11SD16
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊11SD16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C3211SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d3 >

Subgroups: 884 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×12], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C32, C12 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×8], D12 [×12], C2×C12 [×4], C2×C12 [×4], C3×Q8 [×8], C3×Q8 [×4], C22×S3 [×4], C2×SD16, C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×4], C62, C2×C3⋊C8 [×4], Q82S3 [×16], C2×D12 [×4], C6×Q8 [×4], C324C8 [×2], C12⋊S3 [×2], C12⋊S3, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, C22×C3⋊S3, C2×Q82S3 [×4], C2×C324C8, C3211SD16 [×4], C2×C12⋊S3, Q8×C3×C6, C2×C3211SD16
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], SD16 [×2], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C2×SD16, C2×C3⋊S3 [×3], Q82S3 [×8], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C2×Q82S3 [×4], C3211SD16 [×2], C2×C327D4, C2×C3211SD16

Smallest permutation representation of C2×C3211SD16
On 144 points
Generators in S144
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 127)(10 128)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 73)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 89)(40 90)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 142)(114 143)(115 144)(116 137)(117 138)(118 139)(119 140)(120 141)
(1 119 62)(2 63 120)(3 113 64)(4 57 114)(5 115 58)(6 59 116)(7 117 60)(8 61 118)(9 70 50)(10 51 71)(11 72 52)(12 53 65)(13 66 54)(14 55 67)(15 68 56)(16 49 69)(17 121 83)(18 84 122)(19 123 85)(20 86 124)(21 125 87)(22 88 126)(23 127 81)(24 82 128)(25 91 135)(26 136 92)(27 93 129)(28 130 94)(29 95 131)(30 132 96)(31 89 133)(32 134 90)(33 111 74)(34 75 112)(35 105 76)(36 77 106)(37 107 78)(38 79 108)(39 109 80)(40 73 110)(41 137 99)(42 100 138)(43 139 101)(44 102 140)(45 141 103)(46 104 142)(47 143 97)(48 98 144)
(1 22 25)(2 26 23)(3 24 27)(4 28 17)(5 18 29)(6 30 19)(7 20 31)(8 32 21)(9 45 112)(10 105 46)(11 47 106)(12 107 48)(13 41 108)(14 109 42)(15 43 110)(16 111 44)(33 140 69)(34 70 141)(35 142 71)(36 72 143)(37 144 65)(38 66 137)(39 138 67)(40 68 139)(49 74 102)(50 103 75)(51 76 104)(52 97 77)(53 78 98)(54 99 79)(55 80 100)(56 101 73)(57 130 121)(58 122 131)(59 132 123)(60 124 133)(61 134 125)(62 126 135)(63 136 127)(64 128 129)(81 120 92)(82 93 113)(83 114 94)(84 95 115)(85 116 96)(86 89 117)(87 118 90)(88 91 119)
(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)
(2 4)(3 7)(6 8)(9 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)(41 139)(42 142)(43 137)(44 140)(45 143)(46 138)(47 141)(48 144)(49 74)(50 77)(51 80)(52 75)(53 78)(54 73)(55 76)(56 79)(57 120)(58 115)(59 118)(60 113)(61 116)(62 119)(63 114)(64 117)(65 107)(66 110)(67 105)(68 108)(69 111)(70 106)(71 109)(72 112)(81 130)(82 133)(83 136)(84 131)(85 134)(86 129)(87 132)(88 135)(89 128)(90 123)(91 126)(92 121)(93 124)(94 127)(95 122)(96 125)(97 103)(99 101)(100 104)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A 8B 8C 8D 12A ··· 12X order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 36 36 2 2 2 2 2 2 4 4 2 ··· 2 18 18 18 18 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 Q8⋊2S3 kernel C2×C32⋊11SD16 C2×C32⋊4C8 C32⋊11SD16 C2×C12⋊S3 Q8×C3×C6 C6×Q8 C3×C12 C62 C2×C12 C3×Q8 C3×C6 C12 C2×C6 C6 # reps 1 1 4 1 1 4 1 1 4 8 4 8 8 8

Matrix representation of C2×C3211SD16 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 67 67 0 0 0 0 6 67
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,67,6,0,0,0,0,67,67],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C2×C3211SD16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_{11}{\rm SD}_{16}
% in TeX

G:=Group("C2xC3^2:11SD16");
// GroupNames label

G:=SmallGroup(288,798);
// by ID

G=gap.SmallGroup(288,798);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,100,675,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

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