Copied to
clipboard

## G = C2×C4⋊2Q16order 128 = 27

### Direct product of C2 and C4⋊2Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4⋊2Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — C2×C4⋊2Q16
 Lower central C1 — C2 — C2×C4 — C2×C4⋊2Q16
 Upper central C1 — C23 — C2×C42 — C2×C4⋊2Q16
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4⋊2Q16

Generators and relations for C2×C42Q16
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=c-1 >

Subgroups: 396 in 240 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×20], Q8 [×4], Q8 [×18], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×10], C2×Q8 [×13], Q8⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4×Q8 [×4], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16 [×8], C2×Q16 [×8], C22×Q8, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C4⋊C8, C42Q16 [×8], C2×C4×Q8, C2×C4⋊Q8, C22×Q16 [×2], C2×C42Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C8.C22 [×2], C22×D4 [×2], C2×C4○D4, C42Q16 [×4], C2×C4⋊D4, C22×Q16, C2×C8.C22, C2×C42Q16

Smallest permutation representation of C2×C42Q16
Regular action on 128 points
Generators in S128
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 81)(72 82)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 121)(80 122)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 105)(96 106)
(1 82 107 73)(2 74 108 83)(3 84 109 75)(4 76 110 85)(5 86 111 77)(6 78 112 87)(7 88 105 79)(8 80 106 81)(9 26 41 101)(10 102 42 27)(11 28 43 103)(12 104 44 29)(13 30 45 97)(14 98 46 31)(15 32 47 99)(16 100 48 25)(17 51 57 118)(18 119 58 52)(19 53 59 120)(20 113 60 54)(21 55 61 114)(22 115 62 56)(23 49 63 116)(24 117 64 50)(33 66 91 125)(34 126 92 67)(35 68 93 127)(36 128 94 69)(37 70 95 121)(38 122 96 71)(39 72 89 123)(40 124 90 65)
(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 22 5 18)(2 21 6 17)(3 20 7 24)(4 19 8 23)(9 66 13 70)(10 65 14 69)(11 72 15 68)(12 71 16 67)(25 92 29 96)(26 91 30 95)(27 90 31 94)(28 89 32 93)(33 97 37 101)(34 104 38 100)(35 103 39 99)(36 102 40 98)(41 125 45 121)(42 124 46 128)(43 123 47 127)(44 122 48 126)(49 76 53 80)(50 75 54 79)(51 74 55 78)(52 73 56 77)(57 108 61 112)(58 107 62 111)(59 106 63 110)(60 105 64 109)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)

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

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

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

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 D4 Q16 C4○D4 C8.C22 kernel C2×C4⋊2Q16 C2×Q8⋊C4 C2×C4⋊C8 C4⋊2Q16 C2×C4×Q8 C2×C4⋊Q8 C22×Q16 C42 C22×C4 C2×Q8 C2×C4 C2×C4 C22 # reps 1 2 1 8 1 1 2 2 2 4 8 4 2

Matrix representation of C2×C42Q16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 15 0 0 0 0 0 11 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 3 12 0 0 0 0 2 14

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

C2×C42Q16 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes_2Q_{16}
% in TeX

G:=Group("C2xC4:2Q16");
// GroupNames label

G:=SmallGroup(128,1765);
// by ID

G=gap.SmallGroup(128,1765);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,4037,1027,124]);
// 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=c^-1>;
// generators/relations

׿
×
𝔽