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G = C4210Q8order 128 = 27

10th semidirect product of C42 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4210Q8, C42.197D4, C23.560C24, C22.2502- 1+4, C22.3342+ 1+4, C4.9(C4⋊Q8), C429C4.36C2, C428C4.40C2, C424C4.28C2, (C22×C4).859C23, (C2×C42).624C22, C22.372(C22×D4), C22.138(C22×Q8), (C22×Q8).167C22, C2.48(C22.29C24), C23.81C23.29C2, C23.78C23.16C2, C2.C42.274C22, C2.25(C23.41C23), C2.48(C23.38C23), C2.20(C2×C4⋊Q8), (C2×C4⋊Q8).37C2, (C2×C4).406(C2×D4), (C2×C4).135(C2×Q8), (C2×C4⋊C4).383C22, (C2×C42.C2).25C2, SmallGroup(128,1392)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4210Q8
C1C2C22C23C22×C4C2×C4⋊C4C429C4 — C4210Q8
C1C23 — C4210Q8
C1C23 — C4210Q8
C1C23 — C4210Q8

Generators and relations for C4210Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 388 in 224 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], Q8 [×8], C23, C42 [×12], C4⋊C4 [×28], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×8], C2.C42 [×12], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×16], C42.C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C424C4, C428C4 [×2], C429C4 [×2], C23.78C23 [×4], C23.81C23 [×4], C2×C42.C2, C2×C4⋊Q8, C4210Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C2×C4⋊Q8, C22.29C24, C23.38C23, C23.41C23 [×4], C4210Q8

Smallest permutation representation of C4210Q8
Regular action on 128 points
Generators in S128
(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)
(1 124 36 30)(2 121 33 31)(3 122 34 32)(4 123 35 29)(5 41 104 72)(6 42 101 69)(7 43 102 70)(8 44 103 71)(9 55 95 87)(10 56 96 88)(11 53 93 85)(12 54 94 86)(13 83 107 51)(14 84 108 52)(15 81 105 49)(16 82 106 50)(17 78 111 63)(18 79 112 64)(19 80 109 61)(20 77 110 62)(21 59 115 91)(22 60 116 92)(23 57 113 89)(24 58 114 90)(25 67 119 38)(26 68 120 39)(27 65 117 40)(28 66 118 37)(45 126 74 100)(46 127 75 97)(47 128 76 98)(48 125 73 99)
(1 125 117 43)(2 98 118 69)(3 127 119 41)(4 100 120 71)(5 122 46 38)(6 31 47 66)(7 124 48 40)(8 29 45 68)(9 109 50 92)(10 18 51 59)(11 111 52 90)(12 20 49 57)(13 21 56 64)(14 114 53 78)(15 23 54 62)(16 116 55 80)(17 84 58 93)(19 82 60 95)(22 87 61 106)(24 85 63 108)(25 72 34 97)(26 44 35 126)(27 70 36 99)(28 42 33 128)(30 73 65 102)(32 75 67 104)(37 101 121 76)(39 103 123 74)(77 105 113 86)(79 107 115 88)(81 89 94 110)(83 91 96 112)
(1 49 117 12)(2 82 118 95)(3 51 119 10)(4 84 120 93)(5 21 46 64)(6 116 47 80)(7 23 48 62)(8 114 45 78)(9 33 50 28)(11 35 52 26)(13 38 56 122)(14 68 53 29)(15 40 54 124)(16 66 55 31)(17 71 58 100)(18 41 59 127)(19 69 60 98)(20 43 57 125)(22 76 61 101)(24 74 63 103)(25 96 34 83)(27 94 36 81)(30 105 65 86)(32 107 67 88)(37 87 121 106)(39 85 123 108)(42 92 128 109)(44 90 126 111)(70 89 99 110)(72 91 97 112)(73 77 102 113)(75 79 104 115)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4P4Q···4X
order12···244444···44···4
size11···122224···48···8

32 irreducible representations

dim111111112244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2D4Q82+ 1+42- 1+4
kernelC4210Q8C424C4C428C4C429C4C23.78C23C23.81C23C2×C42.C2C2×C4⋊Q8C42C42C22C22
# reps112244114822

Matrix representation of C4210Q8 in GL8(𝔽5)

42000000
41000000
00310000
00020000
00002000
00000200
00001030
00004003
,
10000000
01000000
00100000
00010000
00004300
00001100
00000212
00004344
,
20000000
23000000
00100000
00440000
00001200
00000400
00000343
00000201
,
13000000
14000000
00400000
00040000
00002020
00000033
00001030
00004220

G:=sub<GL(8,GF(5))| [4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,0,4,0,0,0,0,3,1,2,3,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4],[2,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,0,0,2,0,0,0,0,2,3,3,2,0,0,0,0,0,3,0,0] >;

C4210Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{10}Q_8
% in TeX

G:=Group("C4^2:10Q8");
// GroupNames label

G:=SmallGroup(128,1392);
// by ID

G=gap.SmallGroup(128,1392);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,185,80]);
// Polycyclic

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

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