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## G = C2×C16⋊5C4order 128 = 27

### Direct product of C2 and C16⋊5C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C16⋊5C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4×C8 — C2×C16⋊5C4
 Lower central C1 — C2 — C2×C16⋊5C4
 Upper central C1 — C22×C8 — C2×C16⋊5C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C16⋊5C4

Generators and relations for C2×C165C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b9 >

Subgroups: 108 in 100 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C165C4, C2×C4×C8, C22×C16, C2×C165C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, M5(2), C2×C42, C22×C8, C165C4, C2×C4×C8, C2×M5(2), C2×C165C4

Smallest permutation representation of C2×C165C4
Regular action on 128 points
Generators in S128
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(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)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 113)(77 114)(78 115)(79 116)(80 117)(81 110)(82 111)(83 112)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)
(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 91 74 23)(2 84 75 32)(3 93 76 25)(4 86 77 18)(5 95 78 27)(6 88 79 20)(7 81 80 29)(8 90 65 22)(9 83 66 31)(10 92 67 24)(11 85 68 17)(12 94 69 26)(13 87 70 19)(14 96 71 28)(15 89 72 21)(16 82 73 30)(33 101 116 50)(34 110 117 59)(35 103 118 52)(36 112 119 61)(37 105 120 54)(38 98 121 63)(39 107 122 56)(40 100 123 49)(41 109 124 58)(42 102 125 51)(43 111 126 60)(44 104 127 53)(45 97 128 62)(46 106 113 55)(47 99 114 64)(48 108 115 57)

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

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

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

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 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 C8 C8 M5(2) kernel C2×C16⋊5C4 C16⋊5C4 C2×C4×C8 C22×C16 C4×C8 C2×C16 C2×C42 C22×C8 C42 C2×C8 C22×C4 C22 # reps 1 4 1 2 4 16 2 2 8 16 8 16

Matrix representation of C2×C165C4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 9 0 0 0 0 16 0 0 0 0 2 15 0 0 15 15
,
 4 0 0 0 0 16 0 0 0 0 0 16 0 0 1 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,16,0,0,0,0,2,15,0,0,15,15],[4,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;

C2×C165C4 in GAP, Magma, Sage, TeX

C_2\times C_{16}\rtimes_5C_4
% in TeX

G:=Group("C2xC16:5C4");
// GroupNames label

G:=SmallGroup(128,838);
// by ID

G=gap.SmallGroup(128,838);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,925,120,136,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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