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## G = C2×C8⋊4Q8order 128 = 27

### Direct product of C2 and C8⋊4Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C8⋊4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2×C8⋊4Q8
 Lower central C1 — C22 — C2×C8⋊4Q8
 Upper central C1 — C22×C4 — C2×C8⋊4Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊4Q8

Generators and relations for C2×C84Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >

Subgroups: 244 in 200 conjugacy classes, 156 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×20], C2×C4 [×10], Q8 [×8], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C4⋊C8 [×12], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×8], C22×C8 [×2], C22×C8 [×2], C22×Q8, C2×C4×C8, C2×C8⋊C4 [×2], C2×C4⋊C8, C2×C4⋊C8 [×2], C84Q8 [×8], C2×C4×Q8, C2×C84Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C2×M4(2) [×6], C8○D4 [×2], C23×C4, C22×Q8, C2×C4○D4, C84Q8 [×4], C2×C4×Q8, C22×M4(2), C2×C8○D4, C2×C84Q8

Smallest permutation representation of C2×C84Q8
Regular action on 128 points
Generators in S128
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(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 38 43 63)(2 39 44 64)(3 40 45 57)(4 33 46 58)(5 34 47 59)(6 35 48 60)(7 36 41 61)(8 37 42 62)(9 73 18 53)(10 74 19 54)(11 75 20 55)(12 76 21 56)(13 77 22 49)(14 78 23 50)(15 79 24 51)(16 80 17 52)(25 67 118 89)(26 68 119 90)(27 69 120 91)(28 70 113 92)(29 71 114 93)(30 72 115 94)(31 65 116 95)(32 66 117 96)(81 102 107 124)(82 103 108 125)(83 104 109 126)(84 97 110 127)(85 98 111 128)(86 99 112 121)(87 100 105 122)(88 101 106 123)
(1 72 43 94)(2 69 44 91)(3 66 45 96)(4 71 46 93)(5 68 47 90)(6 65 48 95)(7 70 41 92)(8 67 42 89)(9 108 18 82)(10 105 19 87)(11 110 20 84)(12 107 21 81)(13 112 22 86)(14 109 23 83)(15 106 24 88)(16 111 17 85)(25 62 118 37)(26 59 119 34)(27 64 120 39)(28 61 113 36)(29 58 114 33)(30 63 115 38)(31 60 116 35)(32 57 117 40)(49 121 77 99)(50 126 78 104)(51 123 79 101)(52 128 80 98)(53 125 73 103)(54 122 74 100)(55 127 75 97)(56 124 76 102)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 Q8 M4(2) C4○D4 C8○D4 kernel C2×C8⋊4Q8 C2×C4×C8 C2×C8⋊C4 C2×C4⋊C8 C8⋊4Q8 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 C2×C8 C2×C4 C2×C4 C22 # reps 1 1 2 3 8 1 6 8 2 4 8 4 8

Matrix representation of C2×C84Q8 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 4 0
,
 16 0 0 0 0 0 0 9 0 0 0 15 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 15 0 0 0 9 0 0 0 0 0 0 11 9 0 0 0 15 6

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,0,13,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0],[16,0,0,0,0,0,0,15,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,9,0,0,0,15,0,0,0,0,0,0,11,15,0,0,0,9,6] >;

C2×C84Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_4Q_8
% in TeX

G:=Group("C2xC8:4Q8");
// GroupNames label

G:=SmallGroup(128,1691);
// by ID

G=gap.SmallGroup(128,1691);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,268,124]);
// 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^5,d*c*d^-1=c^-1>;
// generators/relations

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