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G = C2×C3211SD16order 288 = 25·32

Direct product of C2 and C3211SD16

direct product, metabelian, supersoluble, monomial

Aliases: C2×C3211SD16, C62.133D4, (C6×Q8)⋊5S3, (C3×Q8)⋊16D6, (C3×C6)⋊11SD16, C63(Q82S3), (C3×C12).101D4, (C2×C12).156D6, C3220(C2×SD16), C12.60(C3⋊D4), C4.8(C327D4), (C6×C12).147C22, (C3×C12).104C23, C12.100(C22×S3), C324C824C22, (Q8×C32)⋊15C22, C12⋊S3.29C22, C22.23(C327D4), (Q8×C3×C6)⋊5C2, Q84(C2×C3⋊S3), (C2×Q8)⋊3(C3⋊S3), C34(C2×Q82S3), (C3×C6).287(C2×D4), C6.128(C2×C3⋊D4), C4.14(C22×C3⋊S3), (C2×C324C8)⋊11C2, (C2×C12⋊S3).15C2, C2.17(C2×C327D4), (C2×C6).101(C3⋊D4), (C2×C4).52(C2×C3⋊S3), SmallGroup(288,798)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C2×C3211SD16
C1C3C32C3×C6C3×C12C12⋊S3C2×C12⋊S3 — C2×C3211SD16
C32C3×C6C3×C12 — C2×C3211SD16
C1C22C2×C4C2×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 2A2B2C2D2E3A3B3C3D4A4B4C4D6A···6L8A8B8C8D12A···12X
order122222333344446···6888812···12
size11113636222222442···2181818184···4

54 irreducible representations

dim11111222222224
type+++++++++++
imageC1C2C2C2C2S3D4D4D6D6SD16C3⋊D4C3⋊D4Q82S3
kernelC2×C3211SD16C2×C324C8C3211SD16C2×C12⋊S3Q8×C3×C6C6×Q8C3×C12C62C2×C12C3×Q8C3×C6C12C2×C6C6
# reps11411411484888

Matrix representation of C2×C3211SD16 in GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
7210000
7200000
00727200
001000
000010
000001
,
7210000
7200000
000100
00727200
000010
000001
,
010000
100000
0072000
001100
00006767
0000667
,
0720000
7200000
001000
00727200
000010
0000072

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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