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G = C32⋊10SD32order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C32⋊10SD32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊5D8 — C32⋊10SD32
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊10SD32
 Upper central C1 — C2 — C4 — C8 — Q16

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 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E ··· 12L 16A 16B 16C 16D 24A ··· 24H order 1 2 2 3 3 3 3 4 4 6 6 6 6 8 8 12 12 12 12 12 ··· 12 16 16 16 16 24 ··· 24 size 1 1 72 2 2 2 2 2 8 2 2 2 2 2 2 4 4 4 4 8 ··· 8 18 18 18 18 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D8 C3⋊D4 SD32 D4⋊S3 C8.6D6 kernel C32⋊10SD32 C24.S3 C32⋊5D8 C32×Q16 C3×Q16 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 4 1 4 2 8 4 4 8

Matrix representation of C3210SD32 in GL6(𝔽97)

 0 1 0 0 0 0 96 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 1 0 0 0 0 96 96 0 0 0 0 0 0 0 1 0 0 0 0 96 96 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 41 56 0 0 0 0 15 56 0 0 0 0 0 0 96 96 0 0 0 0 0 1 0 0 0 0 0 0 54 77 0 0 0 0 10 34
,
 1 1 0 0 0 0 0 96 0 0 0 0 0 0 1 1 0 0 0 0 0 96 0 0 0 0 0 0 1 0 0 0 0 0 1 96

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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