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## G = Q8×C2×C8order 128 = 27

### Direct product of C2×C8 and Q8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C2×C8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 244 in 220 conjugacy classes, 196 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×14], C4 [×6], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×28], C2×C4 [×6], Q8 [×16], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×12], C2×C8 [×6], C22×C4, C22×C4 [×6], C2×Q8 [×12], C4×C8 [×12], C4⋊C8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×C8, C22×C8 [×3], C22×Q8, C2×C4×C8 [×3], C2×C4⋊C8 [×3], C8×Q8 [×8], C2×C4×Q8, Q8×C2×C8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], Q8 [×4], C23 [×15], C2×C8 [×28], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C22×C8 [×14], C8○D4 [×2], C23×C4, C22×Q8, C2×C4○D4, C8×Q8 [×4], C2×C4×Q8, C23×C8, C2×C8○D4, Q8×C2×C8

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

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C8 Q8 C4○D4 C8○D4 kernel Q8×C2×C8 C2×C4×C8 C2×C4⋊C8 C8×Q8 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 C2×Q8 C2×C8 C2×C4 C22 # reps 1 3 3 8 1 6 8 2 32 4 4 8

Matrix representation of Q8×C2×C8 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 2 0 0 0 0 16 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 1 0 0 0 0 1 0 0 0 0 0 4 0 0 4 0
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0] >;

Q8×C2×C8 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_8
% in TeX

G:=Group("Q8xC2xC8");
// GroupNames label

G:=SmallGroup(128,1690);
// by ID

G=gap.SmallGroup(128,1690);
# by ID

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

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