Copied to
clipboard

## G = SD16×C22order 352 = 25·11

### Direct product of C22 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C22
 Chief series C1 — C2 — C4 — C44 — Q8×C11 — C11×SD16 — SD16×C22
 Lower central C1 — C2 — C4 — SD16×C22
 Upper central C1 — C2×C22 — C2×C44 — SD16×C22

Generators and relations for SD16×C22
G = < a,b,c | a22=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C11, C2×C8, SD16, C2×D4, C2×Q8, C22, C22, C22, C2×SD16, C44, C44, C2×C22, C2×C22, C88, C2×C44, C2×C44, D4×C11, D4×C11, Q8×C11, Q8×C11, C22×C22, C2×C88, C11×SD16, D4×C22, Q8×C22, SD16×C22
Quotients: C1, C2, C22, D4, C23, C11, SD16, C2×D4, C22, C2×SD16, C2×C22, D4×C11, C22×C22, C11×SD16, D4×C22, SD16×C22

Smallest permutation representation of SD16×C22
On 176 points
Generators in S176
(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)(133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 154 56 176 101 80 23 122)(2 133 57 155 102 81 24 123)(3 134 58 156 103 82 25 124)(4 135 59 157 104 83 26 125)(5 136 60 158 105 84 27 126)(6 137 61 159 106 85 28 127)(7 138 62 160 107 86 29 128)(8 139 63 161 108 87 30 129)(9 140 64 162 109 88 31 130)(10 141 65 163 110 67 32 131)(11 142 66 164 89 68 33 132)(12 143 45 165 90 69 34 111)(13 144 46 166 91 70 35 112)(14 145 47 167 92 71 36 113)(15 146 48 168 93 72 37 114)(16 147 49 169 94 73 38 115)(17 148 50 170 95 74 39 116)(18 149 51 171 96 75 40 117)(19 150 52 172 97 76 41 118)(20 151 53 173 98 77 42 119)(21 152 54 174 99 78 43 120)(22 153 55 175 100 79 44 121)
(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(67 131)(68 132)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)

G:=sub<Sym(176)| (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,154,56,176,101,80,23,122)(2,133,57,155,102,81,24,123)(3,134,58,156,103,82,25,124)(4,135,59,157,104,83,26,125)(5,136,60,158,105,84,27,126)(6,137,61,159,106,85,28,127)(7,138,62,160,107,86,29,128)(8,139,63,161,108,87,30,129)(9,140,64,162,109,88,31,130)(10,141,65,163,110,67,32,131)(11,142,66,164,89,68,33,132)(12,143,45,165,90,69,34,111)(13,144,46,166,91,70,35,112)(14,145,47,167,92,71,36,113)(15,146,48,168,93,72,37,114)(16,147,49,169,94,73,38,115)(17,148,50,170,95,74,39,116)(18,149,51,171,96,75,40,117)(19,150,52,172,97,76,41,118)(20,151,53,173,98,77,42,119)(21,152,54,174,99,78,43,120)(22,153,55,175,100,79,44,121), (23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(67,131)(68,132)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)>;

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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,154,56,176,101,80,23,122)(2,133,57,155,102,81,24,123)(3,134,58,156,103,82,25,124)(4,135,59,157,104,83,26,125)(5,136,60,158,105,84,27,126)(6,137,61,159,106,85,28,127)(7,138,62,160,107,86,29,128)(8,139,63,161,108,87,30,129)(9,140,64,162,109,88,31,130)(10,141,65,163,110,67,32,131)(11,142,66,164,89,68,33,132)(12,143,45,165,90,69,34,111)(13,144,46,166,91,70,35,112)(14,145,47,167,92,71,36,113)(15,146,48,168,93,72,37,114)(16,147,49,169,94,73,38,115)(17,148,50,170,95,74,39,116)(18,149,51,171,96,75,40,117)(19,150,52,172,97,76,41,118)(20,151,53,173,98,77,42,119)(21,152,54,174,99,78,43,120)(22,153,55,175,100,79,44,121), (23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(67,131)(68,132)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176) );

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),(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,154,56,176,101,80,23,122),(2,133,57,155,102,81,24,123),(3,134,58,156,103,82,25,124),(4,135,59,157,104,83,26,125),(5,136,60,158,105,84,27,126),(6,137,61,159,106,85,28,127),(7,138,62,160,107,86,29,128),(8,139,63,161,108,87,30,129),(9,140,64,162,109,88,31,130),(10,141,65,163,110,67,32,131),(11,142,66,164,89,68,33,132),(12,143,45,165,90,69,34,111),(13,144,46,166,91,70,35,112),(14,145,47,167,92,71,36,113),(15,146,48,168,93,72,37,114),(16,147,49,169,94,73,38,115),(17,148,50,170,95,74,39,116),(18,149,51,171,96,75,40,117),(19,150,52,172,97,76,41,118),(20,151,53,173,98,77,42,119),(21,152,54,174,99,78,43,120),(22,153,55,175,100,79,44,121)], [(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(67,131),(68,132),(69,111),(70,112),(71,113),(72,114),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,121),(80,122),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,129),(88,130),(133,155),(134,156),(135,157),(136,158),(137,159),(138,160),(139,161),(140,162),(141,163),(142,164),(143,165),(144,166),(145,167),(146,168),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176)]])

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 11A ··· 11J 22A ··· 22AD 22AE ··· 22AX 44A ··· 44T 44U ··· 44AN 88A ··· 88AN order 1 2 2 2 2 2 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 1 1 4 4 2 2 4 4 2 2 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C11 C22 C22 C22 C22 D4 D4 SD16 D4×C11 D4×C11 C11×SD16 kernel SD16×C22 C2×C88 C11×SD16 D4×C22 Q8×C22 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C44 C2×C22 C22 C4 C22 C2 # reps 1 1 4 1 1 10 10 40 10 10 1 1 4 10 10 40

Matrix representation of SD16×C22 in GL4(𝔽89) generated by

 25 0 0 0 0 25 0 0 0 0 50 0 0 0 0 50
,
 48 4 0 0 69 41 0 0 0 0 0 40 0 0 69 40
,
 1 0 0 0 65 88 0 0 0 0 1 0 0 0 1 88
G:=sub<GL(4,GF(89))| [25,0,0,0,0,25,0,0,0,0,50,0,0,0,0,50],[48,69,0,0,4,41,0,0,0,0,0,69,0,0,40,40],[1,65,0,0,0,88,0,0,0,0,1,1,0,0,0,88] >;

SD16×C22 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{22}
% in TeX

G:=Group("SD16xC22");
// GroupNames label

G:=SmallGroup(352,168);
// by ID

G=gap.SmallGroup(352,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1056,1081,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c|a^22=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

׿
×
𝔽