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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 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], Q8 [×8], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C8⋊C4 [×4], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C2×C8⋊C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C8⋊Q8 [×8], C2×C42.C2, C2×C4⋊Q8, C2×C8⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C22×Q8 [×2], C8⋊Q8 [×4], 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 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 101)(26 102)(27 103)(28 104)(29 97)(30 98)(31 99)(32 100)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)(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)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)
(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 33)(2 64 47 38)(3 61 48 35)(4 58 41 40)(5 63 42 37)(6 60 43 34)(7 57 44 39)(8 62 45 36)(9 76 22 55)(10 73 23 52)(11 78 24 49)(12 75 17 54)(13 80 18 51)(14 77 19 56)(15 74 20 53)(16 79 21 50)(25 72 120 89)(26 69 113 94)(27 66 114 91)(28 71 115 96)(29 68 116 93)(30 65 117 90)(31 70 118 95)(32 67 119 92)(81 102 111 122)(82 99 112 127)(83 104 105 124)(84 101 106 121)(85 98 107 126)(86 103 108 123)(87 100 109 128)(88 97 110 125)
(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)(9 111 22 81)(10 106 23 84)(11 109 24 87)(12 112 17 82)(13 107 18 85)(14 110 19 88)(15 105 20 83)(16 108 21 86)(25 63 120 37)(26 58 113 40)(27 61 114 35)(28 64 115 38)(29 59 116 33)(30 62 117 36)(31 57 118 39)(32 60 119 34)(49 128 78 100)(50 123 79 103)(51 126 80 98)(52 121 73 101)(53 124 74 104)(54 127 75 99)(55 122 76 102)(56 125 77 97)

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

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

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

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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