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## G = C2×C8⋊Q8order 128 = 27

### Direct product of C2 and C8⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C8⋊Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-1 >

Subgroups: 324 in 192 conjugacy classes, 116 normal (18 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, C8⋊C4, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C2×C8⋊C4, C2×C4.Q8, C2×C2.D8, C8⋊Q8, C2×C42.C2, C2×C4⋊Q8, C2×C8⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C8⋊C22, C8.C22, C22×D4, C22×Q8, C8⋊Q8, C2×C4⋊Q8, C2×C8⋊C22, C2×C8.C22, C2×C8⋊Q8

Smallest permutation representation of C2×C8⋊Q8
Regular action on 128 points
Generators in S128
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 105)(72 106)(81 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)(113 126)(114 127)(115 128)(116 121)(117 122)(118 123)(119 124)(120 125)
(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 59 46 35)(2 64 47 40)(3 61 48 37)(4 58 41 34)(5 63 42 39)(6 60 43 36)(7 57 44 33)(8 62 45 38)(9 74 20 51)(10 79 21 56)(11 76 22 53)(12 73 23 50)(13 78 24 55)(14 75 17 52)(15 80 18 49)(16 77 19 54)(25 72 116 89)(26 69 117 94)(27 66 118 91)(28 71 119 96)(29 68 120 93)(30 65 113 90)(31 70 114 95)(32 67 115 92)(81 98 111 122)(82 103 112 127)(83 100 105 124)(84 97 106 121)(85 102 107 126)(86 99 108 123)(87 104 109 128)(88 101 110 125)
(1 93 46 68)(2 96 47 71)(3 91 48 66)(4 94 41 69)(5 89 42 72)(6 92 43 67)(7 95 44 70)(8 90 45 65)(9 105 20 83)(10 108 21 86)(11 111 22 81)(12 106 23 84)(13 109 24 87)(14 112 17 82)(15 107 18 85)(16 110 19 88)(25 63 116 39)(26 58 117 34)(27 61 118 37)(28 64 119 40)(29 59 120 35)(30 62 113 38)(31 57 114 33)(32 60 115 36)(49 126 80 102)(50 121 73 97)(51 124 74 100)(52 127 75 103)(53 122 76 98)(54 125 77 101)(55 128 78 104)(56 123 79 99)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

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

Matrix representation of C2×C8⋊Q8 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0
,
 15 2 0 0 0 0 0 0 6 2 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 6 9 4 0 0 0 0 11 4 13 9 0 0 0 0 4 8 13 11 0 0 0 0 9 4 6 13
,
 12 15 0 0 0 0 0 0 13 5 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 12 5 0 0 0 0 0 0 0 0 3 7 2 11 0 0 0 0 7 14 11 15 0 0 0 0 15 6 10 3 0 0 0 0 6 2 3 7

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[15,6,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,11,4,9,0,0,0,0,6,4,8,4,0,0,0,0,9,13,13,6,0,0,0,0,4,9,11,13],[12,13,0,0,0,0,0,0,15,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,3,7,15,6,0,0,0,0,7,14,6,2,0,0,0,0,2,11,10,3,0,0,0,0,11,15,3,7] >;

C2×C8⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes Q_8
% in TeX

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

G:=SmallGroup(128,1893);
// by ID

G=gap.SmallGroup(128,1893);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,184,4037,124]);
// 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,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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