Copied to
clipboard

## G = C2×Q16⋊C4order 128 = 27

### Direct product of C2 and Q16⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×Q16⋊C4
G = < a,b,c,d | a2=b8=d4=1, c2=b4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, cd=dc >

Subgroups: 348 in 240 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×16], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×24], Q8 [×8], Q8 [×12], C23, C42 [×4], C42 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], Q16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×12], C2×Q8 [×6], C8⋊C4 [×4], Q8⋊C4 [×8], C4.Q8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×Q8 [×8], C4×Q8 [×4], C22×C8 [×2], C2×Q16 [×12], C22×Q8 [×2], C2×C8⋊C4, C2×Q8⋊C4 [×2], C2×C4.Q8, Q16⋊C4 [×8], C2×C4×Q8 [×2], C22×Q16, C2×Q16⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8.C22 [×4], C23×C4, C22×D4, C2×C4○D4, Q16⋊C4 [×4], C2×C4×D4, C2×C8.C22 [×2], C2×Q16⋊C4

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4AB 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8.C22 kernel C2×Q16⋊C4 C2×C8⋊C4 C2×Q8⋊C4 C2×C4.Q8 Q16⋊C4 C2×C4×Q8 C22×Q16 C2×Q16 C42 C22×C4 C2×C4 C22 # reps 1 1 2 1 8 2 1 16 2 2 4 4

Matrix representation of C2×Q16⋊C4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 10 5 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 15 14 0 0 0 0 0 0 13 2 0 0 0 0 0 0 0 0 16 4 0 0 0 0 0 0 3 1 0 0 0 0 0 0 3 0 13 16 0 0 0 0 2 12 12 4
,
 1 2 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 14 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 3 0
,
 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 11 0 1 2 0 0 0 0 6 6 16 16

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[10,7,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,0,15,13,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,16,3,3,2,0,0,0,0,4,1,0,12,0,0,0,0,0,0,13,12,0,0,0,0,0,0,16,4],[1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,14,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,11,6,0,0,0,0,2,16,0,6,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16] >;

C2×Q16⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xQ16:C4");
// GroupNames label

G:=SmallGroup(128,1673);
// by ID

G=gap.SmallGroup(128,1673);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,1430,184,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations

׿
×
𝔽