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G = C4○D4×C17order 272 = 24·17

Direct product of C17 and C4○D4

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

Aliases: C4○D4×C17, D42C34, Q82C34, C34.13C23, C68.21C22, (C2×C4)⋊3C34, (C2×C68)⋊7C2, (D4×C17)⋊5C2, C4.5(C2×C34), (Q8×C17)⋊5C2, C22.(C2×C34), (C2×C34).2C22, C2.3(C22×C34), SmallGroup(272,49)

Series: Derived Chief Lower central Upper central

C1C2 — C4○D4×C17
C1C2C34C2×C34D4×C17 — C4○D4×C17
C1C2 — C4○D4×C17
C1C68 — C4○D4×C17

Generators and relations for C4○D4×C17
 G = < a,b,c,d | a17=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

2C2
2C2
2C2
2C34
2C34
2C34

Smallest permutation representation of C4○D4×C17
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 41 83 18)(2 42 84 19)(3 43 85 20)(4 44 69 21)(5 45 70 22)(6 46 71 23)(7 47 72 24)(8 48 73 25)(9 49 74 26)(10 50 75 27)(11 51 76 28)(12 35 77 29)(13 36 78 30)(14 37 79 31)(15 38 80 32)(16 39 81 33)(17 40 82 34)(52 110 135 96)(53 111 136 97)(54 112 120 98)(55 113 121 99)(56 114 122 100)(57 115 123 101)(58 116 124 102)(59 117 125 86)(60 118 126 87)(61 119 127 88)(62 103 128 89)(63 104 129 90)(64 105 130 91)(65 106 131 92)(66 107 132 93)(67 108 133 94)(68 109 134 95)
(1 18 83 41)(2 19 84 42)(3 20 85 43)(4 21 69 44)(5 22 70 45)(6 23 71 46)(7 24 72 47)(8 25 73 48)(9 26 74 49)(10 27 75 50)(11 28 76 51)(12 29 77 35)(13 30 78 36)(14 31 79 37)(15 32 80 38)(16 33 81 39)(17 34 82 40)(52 110 135 96)(53 111 136 97)(54 112 120 98)(55 113 121 99)(56 114 122 100)(57 115 123 101)(58 116 124 102)(59 117 125 86)(60 118 126 87)(61 119 127 88)(62 103 128 89)(63 104 129 90)(64 105 130 91)(65 106 131 92)(66 107 132 93)(67 108 133 94)(68 109 134 95)
(1 133)(2 134)(3 135)(4 136)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 103)(31 104)(32 105)(33 106)(34 107)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 86)(51 87)(52 85)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)

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

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

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

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++++
imageC1C2C2C2C17C34C34C34C4○D4C4○D4×C17
kernelC4○D4×C17C2×C68D4×C17Q8×C17C4○D4C2×C4D4Q8C17C1
# reps133116484816232

Matrix representation of C4○D4×C17 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] >;

C4○D4×C17 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 C4○D4×C17 in TeX

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