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G = C4oD4xC17order 272 = 24·17

Direct product of C17 and C4oD4

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

Aliases: C4oD4xC17, D4:2C34, Q8:2C34, C34.13C23, C68.21C22, (C2xC4):3C34, (C2xC68):7C2, (D4xC17):5C2, C4.5(C2xC34), (Q8xC17):5C2, C22.(C2xC34), (C2xC34).2C22, C2.3(C22xC34), SmallGroup(272,49)

Series: Derived Chief Lower central Upper central

C1C2 — C4oD4xC17
C1C2C34C2xC34D4xC17 — C4oD4xC17
C1C2 — C4oD4xC17
C1C68 — C4oD4xC17

Generators and relations for C4oD4xC17
 G = < a,b,c,d | a17=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 46 in 40 conjugacy classes, 34 normal (10 characteristic)
Quotients: C1, C2, C22, C23, C4oD4, C17, C34, C2xC34, C22xC34, C4oD4xC17
2C2
2C2
2C2
2C34
2C34
2C34

Smallest permutation representation of C4oD4xC17
On 136 points
Generators in S136
(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)
(1 58 33 135)(2 59 34 136)(3 60 18 120)(4 61 19 121)(5 62 20 122)(6 63 21 123)(7 64 22 124)(8 65 23 125)(9 66 24 126)(10 67 25 127)(11 68 26 128)(12 52 27 129)(13 53 28 130)(14 54 29 131)(15 55 30 132)(16 56 31 133)(17 57 32 134)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 135 33 58)(2 136 34 59)(3 120 18 60)(4 121 19 61)(5 122 20 62)(6 123 21 63)(7 124 22 64)(8 125 23 65)(9 126 24 66)(10 127 25 67)(11 128 26 68)(12 129 27 52)(13 130 28 53)(14 131 29 54)(15 132 30 55)(16 133 31 56)(17 134 32 57)(35 111 102 77)(36 112 86 78)(37 113 87 79)(38 114 88 80)(39 115 89 81)(40 116 90 82)(41 117 91 83)(42 118 92 84)(43 119 93 85)(44 103 94 69)(45 104 95 70)(46 105 96 71)(47 106 97 72)(48 107 98 73)(49 108 99 74)(50 109 100 75)(51 110 101 76)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 69)(34 70)(35 126)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 86)(68 87)

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

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

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

170 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E17A···17P34A···34P34Q···34BL68A···68AF68AG···68CB
order122224444417···1734···3434···3468···6868···68
size11222112221···11···12···21···12···2

170 irreducible representations

dim1111111122
type++++
imageC1C2C2C2C17C34C34C34C4oD4C4oD4xC17
kernelC4oD4xC17C2xC68D4xC17Q8xC17C4oD4C2xC4D4Q8C17C1
# reps133116484816232

Matrix representation of C4oD4xC17 in GL2(F137) generated by

1230
0123
,
1000
0100
,
3746
0100
,
46123
6391
G:=sub<GL(2,GF(137))| [123,0,0,123],[100,0,0,100],[37,0,46,100],[46,63,123,91] >;

C4oD4xC17 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{17}
% in TeX

G:=Group("C4oD4xC17");
// GroupNames label

G:=SmallGroup(272,49);
// by ID

G=gap.SmallGroup(272,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-17,-2,1381,522]);
// Polycyclic

G:=Group<a,b,c,d|a^17=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

Export

Subgroup lattice of C4oD4xC17 in TeX

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