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G = C2×C83Q8order 128 = 27

Direct product of C2 and C83Q8

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

Aliases: C2×C83Q8, C42.361D4, C42.715C23, C86(C2×Q8), (C2×C8)⋊14Q8, C4.12(C4⋊Q8), C4.8(C22×Q8), C4⋊C4.96C23, (C2×C4).81SD16, C4.22(C2×SD16), (C2×C8).596C23, (C4×C8).430C22, (C2×C4).355C24, C23.882(C2×D4), (C22×C4).615D4, C4⋊Q8.282C22, C22.44(C4⋊Q8), C22.89(C2×SD16), C2.19(C22×SD16), C4.Q8.157C22, (C22×C8).567C22, C22.615(C22×D4), (C22×C4).1564C23, (C2×C42).1130C22, (C2×C4×C8).56C2, C2.25(C2×C4⋊Q8), (C2×C4⋊Q8).49C2, (C2×C4).860(C2×D4), (C2×C4).244(C2×Q8), (C2×C4.Q8).33C2, (C2×C4⋊C4).628C22, SmallGroup(128,1889)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C83Q8
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C2×C83Q8
C1C2C2×C4 — C2×C83Q8
C1C23C2×C42 — C2×C83Q8
C1C2C2C2×C4 — C2×C83Q8

Generators and relations for C2×C83Q8
 G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 372 in 212 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×16], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], C4×C8 [×4], C4.Q8 [×16], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C2×C4×C8, C2×C4.Q8 [×4], C83Q8 [×8], C2×C4⋊Q8 [×2], C2×C83Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], SD16 [×8], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C2×SD16 [×12], C22×D4, C22×Q8 [×2], C83Q8 [×4], C2×C4⋊Q8, C22×SD16 [×2], C2×C83Q8

Smallest permutation representation of C2×C83Q8
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 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 102)(26 103)(27 104)(28 97)(29 98)(30 99)(31 100)(32 101)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(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 63 47 40)(2 64 48 33)(3 57 41 34)(4 58 42 35)(5 59 43 36)(6 60 44 37)(7 61 45 38)(8 62 46 39)(9 78 24 53)(10 79 17 54)(11 80 18 55)(12 73 19 56)(13 74 20 49)(14 75 21 50)(15 76 22 51)(16 77 23 52)(25 66 120 93)(26 67 113 94)(27 68 114 95)(28 69 115 96)(29 70 116 89)(30 71 117 90)(31 72 118 91)(32 65 119 92)(81 99 108 126)(82 100 109 127)(83 101 110 128)(84 102 111 121)(85 103 112 122)(86 104 105 123)(87 97 106 124)(88 98 107 125)
(1 93 47 66)(2 96 48 69)(3 91 41 72)(4 94 42 67)(5 89 43 70)(6 92 44 65)(7 95 45 68)(8 90 46 71)(9 110 24 83)(10 105 17 86)(11 108 18 81)(12 111 19 84)(13 106 20 87)(14 109 21 82)(15 112 22 85)(16 107 23 88)(25 63 120 40)(26 58 113 35)(27 61 114 38)(28 64 115 33)(29 59 116 36)(30 62 117 39)(31 57 118 34)(32 60 119 37)(49 124 74 97)(50 127 75 100)(51 122 76 103)(52 125 77 98)(53 128 78 101)(54 123 79 104)(55 126 80 99)(56 121 73 102)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim111112222
type++++++-+
imageC1C2C2C2C2D4Q8D4SD16
kernelC2×C83Q8C2×C4×C8C2×C4.Q8C83Q8C2×C4⋊Q8C42C2×C8C22×C4C2×C4
# reps1148228216

Matrix representation of C2×C83Q8 in GL5(𝔽17)

160000
016000
001600
000160
000016
,
160000
012500
0121200
000125
0001212
,
10000
001600
01000
00010
00001
,
160000
05500
051200
000411
0001113

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],[16,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,12,12,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,4,11,0,0,0,11,13] >;

C2×C83Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_3Q_8
% in TeX

G:=Group("C2xC8:3Q8");
// GroupNames label

G:=SmallGroup(128,1889);
// by ID

G=gap.SmallGroup(128,1889);
# by ID

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

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