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## G = C2×C4⋊C16order 128 = 27

### Direct product of C2 and C4⋊C16

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C4⋊C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C4×C8 — C2×C4⋊C16
 Lower central C1 — C2 — C2×C4⋊C16
 Upper central C1 — C22×C8 — C2×C4⋊C16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C4⋊C16

Generators and relations for C2×C4⋊C16
G = < a,b,c | a2=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 108 in 92 conjugacy classes, 76 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×2], C23, C16 [×4], C42 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4 [×3], C4×C8 [×4], C2×C16 [×4], C2×C16 [×4], C2×C42, C22×C8 [×2], C4⋊C16 [×4], C2×C4×C8, C22×C16 [×2], C2×C4⋊C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C16 [×4], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C16 [×6], M5(2) [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C4⋊C16 [×4], C2×C4⋊C8, C22×C16, C2×M5(2), C2×C4⋊C16

Smallest permutation representation of C2×C4⋊C16
Regular action on 128 points
Generators in S128
(1 108)(2 109)(3 110)(4 111)(5 112)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 63)(18 64)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
(1 25 67 95)(2 96 68 26)(3 27 69 81)(4 82 70 28)(5 29 71 83)(6 84 72 30)(7 31 73 85)(8 86 74 32)(9 17 75 87)(10 88 76 18)(11 19 77 89)(12 90 78 20)(13 21 79 91)(14 92 80 22)(15 23 65 93)(16 94 66 24)(33 102 49 115)(34 116 50 103)(35 104 51 117)(36 118 52 105)(37 106 53 119)(38 120 54 107)(39 108 55 121)(40 122 56 109)(41 110 57 123)(42 124 58 111)(43 112 59 125)(44 126 60 97)(45 98 61 127)(46 128 62 99)(47 100 63 113)(48 114 64 101)
(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)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 8A ··· 8P 8Q ··· 8X 16A ··· 16AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 C8 C16 D4 Q8 M4(2) M5(2) kernel C2×C4⋊C16 C4⋊C16 C2×C4×C8 C22×C16 C4×C8 C2×C42 C22×C8 C42 C22×C4 C2×C4 C2×C8 C2×C8 C2×C4 C22 # reps 1 4 1 2 4 2 2 8 8 32 2 2 4 8

Matrix representation of C2×C4⋊C16 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 4 0 0 0 0 13
,
 14 0 0 0 0 13 0 0 0 0 0 1 0 0 9 0
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[14,0,0,0,0,13,0,0,0,0,0,9,0,0,1,0] >;

C2×C4⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4:C16");
// GroupNames label

G:=SmallGroup(128,881);
// by ID

G=gap.SmallGroup(128,881);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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