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## G = C2×C8⋊2Q8order 128 = 27

### Direct product of C2 and C8⋊2Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2×C8⋊2Q8
 Lower central C1 — C2 — C2×C4 — C2×C8⋊2Q8
 Upper central C1 — C23 — C2×C42 — C2×C8⋊2Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊2Q8

Generators and relations for C2×C82Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 372 in 212 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C4×C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C2×C4×C8, C2×C2.D8, C82Q8, C2×C4⋊Q8, C2×C82Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, C24, C4⋊Q8, C2×D8, C2×Q16, C22×D4, C22×Q8, C82Q8, C2×C4⋊Q8, C22×D8, C22×Q16, C2×C82Q8

Smallest permutation representation of C2×C82Q8
Regular action on 128 points
Generators in S128
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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 59 43 36)(2 60 44 37)(3 61 45 38)(4 62 46 39)(5 63 47 40)(6 64 48 33)(7 57 41 34)(8 58 42 35)(9 78 24 53)(10 79 17 54)(11 80 18 55)(12 73 19 56)(13 74 20 49)(14 75 21 50)(15 76 22 51)(16 77 23 52)(25 67 115 94)(26 68 116 95)(27 69 117 96)(28 70 118 89)(29 71 119 90)(30 72 120 91)(31 65 113 92)(32 66 114 93)(81 97 106 124)(82 98 107 125)(83 99 108 126)(84 100 109 127)(85 101 110 128)(86 102 111 121)(87 103 112 122)(88 104 105 123)
(1 93 43 66)(2 92 44 65)(3 91 45 72)(4 90 46 71)(5 89 47 70)(6 96 48 69)(7 95 41 68)(8 94 42 67)(9 110 24 85)(10 109 17 84)(11 108 18 83)(12 107 19 82)(13 106 20 81)(14 105 21 88)(15 112 22 87)(16 111 23 86)(25 58 115 35)(26 57 116 34)(27 64 117 33)(28 63 118 40)(29 62 119 39)(30 61 120 38)(31 60 113 37)(32 59 114 36)(49 124 74 97)(50 123 75 104)(51 122 76 103)(52 121 77 102)(53 128 78 101)(54 127 79 100)(55 126 80 99)(56 125 73 98)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 D4 Q8 D4 D8 Q16 kernel C2×C8⋊2Q8 C2×C4×C8 C2×C2.D8 C8⋊2Q8 C2×C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C2×C4 # reps 1 1 4 8 2 2 8 2 8 8

Matrix representation of C2×C82Q8 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 15 0 0 0 1 1 0 0 0 0 0 3 14 0 0 0 3 3
,
 1 0 0 0 0 0 16 15 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 9 3 0 0 0 1 8 0 0 0 0 0 5 5 0 0 0 5 12

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,9,1,0,0,0,3,8,0,0,0,0,0,5,5,0,0,0,5,12] >;

C2×C82Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2Q_8
% in TeX

G:=Group("C2xC8:2Q8");
// GroupNames label

G:=SmallGroup(128,1891);
// by ID

G=gap.SmallGroup(128,1891);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,4037,124]);
// Polycyclic

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

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