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G = C2×C8.5Q8order 128 = 27

Direct product of C2 and C8.5Q8

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

Aliases: C2×C8.5Q8, C42.362D4, C42.716C23, C8.22(C2×Q8), (C2×C8).47Q8, C4.13(C4⋊Q8), C4.9(C22×Q8), C4⋊C4.97C23, (C2×C4).356C24, (C2×C8).562C23, (C4×C8).417C22, (C22×C4).567D4, C23.883(C2×D4), C22.45(C4⋊Q8), C22.99(C4○D8), C4.Q8.158C22, C2.D8.177C22, (C22×C8).568C22, C22.616(C22×D4), (C2×C42).1131C22, (C22×C4).1565C23, C42.C2.114C22, (C2×C4×C8).47C2, C2.26(C2×C4⋊Q8), C2.32(C2×C4○D8), (C2×C4).696(C2×D4), (C2×C4).245(C2×Q8), (C2×C2.D8).29C2, (C2×C4.Q8).34C2, (C2×C4⋊C4).629C22, (C2×C42.C2).33C2, SmallGroup(128,1890)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C8.5Q8
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C2×C8.5Q8
C1C2C2×C4 — C2×C8.5Q8
C1C23C2×C42 — C2×C8.5Q8
C1C2C2C2×C4 — C2×C8.5Q8

Generators and relations for C2×C8.5Q8
 G = < a,b,c,d | a2=b8=c4=1, d2=b4c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >

Subgroups: 276 in 180 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×20], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C4×C8 [×4], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C42.C2 [×8], C42.C2 [×4], C22×C8 [×2], C2×C4×C8, C2×C4.Q8 [×2], C2×C2.D8 [×2], C8.5Q8 [×8], C2×C42.C2 [×2], C2×C8.5Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C4○D8 [×4], C22×D4, C22×Q8 [×2], C8.5Q8 [×4], C2×C4⋊Q8, C2×C4○D8 [×2], C2×C8.5Q8

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

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim1111112222
type+++++++-+
imageC1C2C2C2C2C2D4Q8D4C4○D8
kernelC2×C8.5Q8C2×C4×C8C2×C4.Q8C2×C2.D8C8.5Q8C2×C42.C2C42C2×C8C22×C4C22
# reps11228228216

Matrix representation of C2×C8.5Q8 in GL5(𝔽17)

160000
016000
001600
000160
000016
,
160000
016000
001600
000512
00055
,
10000
00100
016000
000130
000013
,
160000
0121200
012500
00010
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,5,5,0,0,0,12,5],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,12,12,0,0,0,12,5,0,0,0,0,0,1,0,0,0,0,0,16] >;

C2×C8.5Q8 in GAP, Magma, Sage, TeX

C_2\times C_8._5Q_8
% in TeX

G:=Group("C2xC8.5Q8");
// GroupNames label

G:=SmallGroup(128,1890);
// by ID

G=gap.SmallGroup(128,1890);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,184,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations

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