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## G = C2×C4.Q16order 128 = 27

### Direct product of C2 and C4.Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4.Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — C2×C4.Q16
 Lower central C1 — C2 — C2×C4 — C2×C4.Q16
 Upper central C1 — C23 — C2×C42 — C2×C4.Q16
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4.Q16

Generators and relations for C2×C4.Q16
G = < a,b,c,d | a2=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >

Subgroups: 348 in 208 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C22×Q8, C2×Q8⋊C4, C2×C4⋊C8, C2×C2.D8, C4.Q16, C2×C4×Q8, C2×C4⋊Q8, C2×C4.Q16
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C2×Q16, C8⋊C22, C22×D4, C22×Q8, C2×C4○D4, C4.Q16, C2×C22⋊Q8, C22×Q16, C2×C8⋊C22, C2×C4.Q16

Smallest permutation representation of C2×C4.Q16
Regular action on 128 points
Generators in S128
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 121)(24 122)(25 100)(26 101)(27 102)(28 103)(29 104)(30 97)(31 98)(32 99)(33 93)(34 94)(35 95)(36 96)(37 89)(38 90)(39 91)(40 92)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)(81 110)(82 111)(83 112)(84 105)(85 106)(86 107)(87 108)(88 109)
(1 21 95 86)(2 87 96 22)(3 23 89 88)(4 81 90 24)(5 17 91 82)(6 83 92 18)(7 19 93 84)(8 85 94 20)(9 70 41 101)(10 102 42 71)(11 72 43 103)(12 104 44 65)(13 66 45 97)(14 98 46 67)(15 68 47 99)(16 100 48 69)(25 80 59 120)(26 113 60 73)(27 74 61 114)(28 115 62 75)(29 76 63 116)(30 117 64 77)(31 78 57 118)(32 119 58 79)(33 105 54 125)(34 126 55 106)(35 107 56 127)(36 128 49 108)(37 109 50 121)(38 122 51 110)(39 111 52 123)(40 124 53 112)
(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)
(1 62 5 58)(2 27 6 31)(3 60 7 64)(4 25 8 29)(9 105 13 109)(10 124 14 128)(11 111 15 107)(12 122 16 126)(17 79 21 75)(18 118 22 114)(19 77 23 73)(20 116 24 120)(26 93 30 89)(28 91 32 95)(33 97 37 101)(34 65 38 69)(35 103 39 99)(36 71 40 67)(41 125 45 121)(42 112 46 108)(43 123 47 127)(44 110 48 106)(49 102 53 98)(50 70 54 66)(51 100 55 104)(52 68 56 72)(57 96 61 92)(59 94 63 90)(74 83 78 87)(76 81 80 85)(82 119 86 115)(84 117 88 113)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 D4 D4 Q8 Q16 C4○D4 C8⋊C22 kernel C2×C4.Q16 C2×Q8⋊C4 C2×C4⋊C8 C2×C2.D8 C4.Q16 C2×C4×Q8 C2×C4⋊Q8 C42 C22×C4 C2×Q8 C2×C4 C2×C4 C22 # reps 1 2 1 2 8 1 1 2 2 4 8 4 2

Matrix representation of C2×C4.Q16 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 13 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 16 0 0 0 16 0 0 0 0 0 0 11 11 0 0 0 3 0
,
 1 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 6 15 0 0 0 10 11

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,11,3,0,0,0,11,0],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,6,10,0,0,0,15,11] >;

C2×C4.Q16 in GAP, Magma, Sage, TeX

C_2\times C_4.Q_{16}
% in TeX

G:=Group("C2xC4.Q16");
// GroupNames label

G:=SmallGroup(128,1806);
// by ID

G=gap.SmallGroup(128,1806);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations

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