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## G = C2×C22.4Q16order 128 = 27

### Direct product of C2 and C22.4Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C22.4Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C22.4Q16
 Lower central C1 — C2 — C4 — C2×C22.4Q16
 Upper central C1 — C24 — C23×C4 — C2×C22.4Q16
 Jennings C1 — C2 — C2 — C22×C4 — C2×C22.4Q16

Generators and relations for C2×C22.4Q16
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=cd4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 436 in 256 conjugacy classes, 156 normal (18 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×8], C22 [×3], C22 [×32], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×32], C23, C23 [×14], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×20], C24, C2×C4⋊C4 [×12], C2×C4⋊C4 [×6], C22×C8 [×6], C22×C8 [×4], C23×C4, C23×C4 [×2], C22.4Q16 [×4], C22×C4⋊C4 [×2], C23×C8, C2×C22.4Q16
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], D4⋊C4 [×8], Q8⋊C4 [×8], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×D8, C2×SD16 [×2], C2×Q16, C22.4Q16 [×8], C2×C2.C42, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C2×C4.Q8, C2×C2.D8, C2×C22.4Q16

Smallest permutation representation of C2×C22.4Q16
Regular action on 128 points
Generators in S128
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 92)(34 93)(35 94)(36 95)(37 96)(38 89)(39 90)(40 91)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 73)(56 74)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 81)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)(105 126)(106 127)(107 128)(108 121)(109 122)(110 123)(111 124)(112 125)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 94)(58 95)(59 96)(60 89)(61 90)(62 91)(63 92)(64 93)(65 119)(66 120)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 97)(88 98)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 104)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)
(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 67 63 112)(2 120 64 76)(3 65 57 110)(4 118 58 74)(5 71 59 108)(6 116 60 80)(7 69 61 106)(8 114 62 78)(9 84 26 125)(10 101 27 50)(11 82 28 123)(12 99 29 56)(13 88 30 121)(14 97 31 54)(15 86 32 127)(16 103 25 52)(17 115 90 79)(18 68 91 105)(19 113 92 77)(20 66 93 111)(21 119 94 75)(22 72 95 109)(23 117 96 73)(24 70 89 107)(33 51 43 102)(34 124 44 83)(35 49 45 100)(36 122 46 81)(37 55 47 98)(38 128 48 87)(39 53 41 104)(40 126 42 85)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 D4 Q8 D4 D8 SD16 Q16 kernel C2×C22.4Q16 C22.4Q16 C22×C4⋊C4 C23×C8 C2×C4⋊C4 C22×C8 C22×C4 C22×C4 C24 C23 C23 C23 # reps 1 4 2 1 16 8 5 2 1 4 8 4

Matrix representation of C2×C22.4Q16 in GL6(𝔽17)

 1 0 0 0 0 0 0 16 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
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 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
,
 13 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 11 0 0 0 0 3 6
,
 4 0 0 0 0 0 0 16 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 7 7 0 0 0 0 5 10

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

C2×C22.4Q16 in GAP, Magma, Sage, TeX

C_2\times C_2^2._4Q_{16}
% in TeX

G:=Group("C2xC2^2.4Q16");
// GroupNames label

G:=SmallGroup(128,466);
// by ID

G=gap.SmallGroup(128,466);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,4037,124]);
// Polycyclic

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

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