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G = C3210SD32order 288 = 25·32

2nd semidirect product of C32 and SD32 acting via SD32/Q16=C2

metabelian, supersoluble, monomial

Aliases: C24.20D6, C3210SD32, (C3×Q16)⋊1S3, (C3×C6).39D8, Q161(C3⋊S3), (C3×C12).54D4, C24.S35C2, C325D8.3C2, C6.25(D4⋊S3), C33(C8.6D6), (C32×Q16)⋊2C2, C12.36(C3⋊D4), (C3×C24).19C22, C2.6(C327D8), C4.3(C327D4), C8.6(C2×C3⋊S3), SmallGroup(288,303)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C3210SD32
C1C3C32C3×C6C3×C12C3×C24C325D8 — C3210SD32
C32C3×C6C3×C12C3×C24 — C3210SD32
C1C2C4C8Q16

Generators and relations for C3210SD32
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c7 >

Subgroups: 440 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, D4, Q8, C32, C12 [×4], C12 [×4], D6 [×4], C16, D8, Q16, C3⋊S3, C3×C6, C24 [×4], D12 [×4], C3×Q8 [×4], SD32, C3×C12, C3×C12, C2×C3⋊S3, C3⋊C16 [×4], D24 [×4], C3×Q16 [×4], C3×C24, C12⋊S3, Q8×C32, C8.6D6 [×4], C24.S3, C325D8, C32×Q16, C3210SD32
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], SD32, C2×C3⋊S3, D4⋊S3 [×4], C327D4, C8.6D6 [×4], C327D8, C3210SD32

Smallest permutation representation of C3210SD32
On 144 points
Generators in S144
(1 104 22)(2 23 105)(3 106 24)(4 25 107)(5 108 26)(6 27 109)(7 110 28)(8 29 111)(9 112 30)(10 31 97)(11 98 32)(12 17 99)(13 100 18)(14 19 101)(15 102 20)(16 21 103)(33 67 92)(34 93 68)(35 69 94)(36 95 70)(37 71 96)(38 81 72)(39 73 82)(40 83 74)(41 75 84)(42 85 76)(43 77 86)(44 87 78)(45 79 88)(46 89 80)(47 65 90)(48 91 66)(49 136 121)(50 122 137)(51 138 123)(52 124 139)(53 140 125)(54 126 141)(55 142 127)(56 128 143)(57 144 113)(58 114 129)(59 130 115)(60 116 131)(61 132 117)(62 118 133)(63 134 119)(64 120 135)
(1 78 124)(2 125 79)(3 80 126)(4 127 65)(5 66 128)(6 113 67)(7 68 114)(8 115 69)(9 70 116)(10 117 71)(11 72 118)(12 119 73)(13 74 120)(14 121 75)(15 76 122)(16 123 77)(17 63 82)(18 83 64)(19 49 84)(20 85 50)(21 51 86)(22 87 52)(23 53 88)(24 89 54)(25 55 90)(26 91 56)(27 57 92)(28 93 58)(29 59 94)(30 95 60)(31 61 96)(32 81 62)(33 109 144)(34 129 110)(35 111 130)(36 131 112)(37 97 132)(38 133 98)(39 99 134)(40 135 100)(41 101 136)(42 137 102)(43 103 138)(44 139 104)(45 105 140)(46 141 106)(47 107 142)(48 143 108)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 101)(18 108)(19 99)(20 106)(21 97)(22 104)(23 111)(24 102)(25 109)(26 100)(27 107)(28 98)(29 105)(30 112)(31 103)(32 110)(33 55)(34 62)(35 53)(36 60)(37 51)(38 58)(39 49)(40 56)(41 63)(42 54)(43 61)(44 52)(45 59)(46 50)(47 57)(48 64)(65 113)(66 120)(67 127)(68 118)(69 125)(70 116)(71 123)(72 114)(73 121)(74 128)(75 119)(76 126)(77 117)(78 124)(79 115)(80 122)(81 129)(82 136)(83 143)(84 134)(85 141)(86 132)(87 139)(88 130)(89 137)(90 144)(91 135)(92 142)(93 133)(94 140)(95 131)(96 138)

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

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

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

39 conjugacy classes

class 1 2A2B3A3B3C3D4A4B6A6B6C6D8A8B12A12B12C12D12E···12L16A16B16C16D24A···24H
order1223333446666881212121212···121616161624···24
size117222222822222244448···8181818184···4

39 irreducible representations

dim111122222244
type++++++++++
imageC1C2C2C2S3D4D6D8C3⋊D4SD32D4⋊S3C8.6D6
kernelC3210SD32C24.S3C325D8C32×Q16C3×Q16C3×C12C24C3×C6C12C32C6C3
# reps111141428448

Matrix representation of C3210SD32 in GL6(𝔽97)

010000
96960000
001000
000100
000010
000001
,
010000
96960000
000100
00969600
000010
000001
,
41560000
15560000
00969600
000100
00005477
00001034
,
110000
0960000
001100
0009600
000010
0000196

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,96,0,0,0,0,1,96,0,0,0,0,0,0,0,96,0,0,0,0,1,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[41,15,0,0,0,0,56,56,0,0,0,0,0,0,96,0,0,0,0,0,96,1,0,0,0,0,0,0,54,10,0,0,0,0,77,34],[1,0,0,0,0,0,1,96,0,0,0,0,0,0,1,0,0,0,0,0,1,96,0,0,0,0,0,0,1,1,0,0,0,0,0,96] >;

C3210SD32 in GAP, Magma, Sage, TeX

C_3^2\rtimes_{10}{\rm SD}_{32}
% in TeX

G:=Group("C3^2:10SD32");
// GroupNames label

G:=SmallGroup(288,303);
// by ID

G=gap.SmallGroup(288,303);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^7>;
// generators/relations

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