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G = C42.37Q8order 128 = 27

37th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.37Q8, C23.471C24, C22.2542+ 1+4, C22.1922- 1+4, C428C4.36C2, C429C4.33C2, C4.33(C42.C2), (C2×C42).570C22, (C22×C4).545C23, C22.112(C22×Q8), C23.83C23.16C2, C23.65C23.57C2, C2.C42.207C22, C2.27(C22.49C24), C2.42(C22.50C24), C2.20(C22.53C24), C2.15(C23.41C23), (C4×C4⋊C4).70C2, (C2×C4).232(C2×Q8), C2.18(C2×C42.C2), (C2×C4).896(C4○D4), (C2×C4⋊C4).318C22, C22.347(C2×C4○D4), SmallGroup(128,1303)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.37Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.37Q8
C1C23 — C42.37Q8
C1C23 — C42.37Q8
C1C23 — C42.37Q8

Generators and relations for C42.37Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 308 in 190 conjugacy classes, 108 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×14], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×38], C23, C42 [×4], C42 [×8], C4⋊C4 [×22], C22×C4, C22×C4 [×14], C2.C42 [×16], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×14], C4×C4⋊C4 [×2], C428C4 [×4], C429C4, C23.65C23 [×4], C23.83C23 [×4], C42.37Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42.C2 [×8], C22×Q8, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C42.C2 [×2], C23.41C23, C22.49C24, C22.50C24 [×2], C22.53C24, C42.37Q8

Smallest permutation representation of C42.37Q8
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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 110 39 87)(6 111 40 88)(7 112 37 85)(8 109 38 86)(9 29 18 58)(10 30 19 59)(11 31 20 60)(12 32 17 57)(21 98 46 65)(22 99 47 66)(23 100 48 67)(24 97 45 68)(25 69 50 64)(26 70 51 61)(27 71 52 62)(28 72 49 63)(33 116 126 83)(34 113 127 84)(35 114 128 81)(36 115 125 82)(73 93 106 120)(74 94 107 117)(75 95 108 118)(76 96 105 119)(77 124 104 91)(78 121 101 92)(79 122 102 89)(80 123 103 90)
(1 46 9 72)(2 24 10 62)(3 48 11 70)(4 22 12 64)(5 76 113 89)(6 108 114 121)(7 74 115 91)(8 106 116 123)(13 68 59 52)(14 100 60 26)(15 66 57 50)(16 98 58 28)(17 69 54 47)(18 63 55 21)(19 71 56 45)(20 61 53 23)(25 44 99 32)(27 42 97 30)(29 49 41 65)(31 51 43 67)(33 80 86 93)(34 102 87 119)(35 78 88 95)(36 104 85 117)(37 107 82 124)(38 73 83 90)(39 105 84 122)(40 75 81 92)(77 112 94 125)(79 110 96 127)(101 111 118 128)(103 109 120 126)
(1 104 11 119)(2 78 12 93)(3 102 9 117)(4 80 10 95)(5 65 115 51)(6 99 116 27)(7 67 113 49)(8 97 114 25)(13 121 57 106)(14 89 58 74)(15 123 59 108)(16 91 60 76)(17 120 56 101)(18 94 53 79)(19 118 54 103)(20 96 55 77)(21 36 61 87)(22 126 62 111)(23 34 63 85)(24 128 64 109)(26 39 98 82)(28 37 100 84)(29 107 43 122)(30 75 44 90)(31 105 41 124)(32 73 42 92)(33 71 88 47)(35 69 86 45)(38 68 81 50)(40 66 83 52)(46 125 70 110)(48 127 72 112)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC42.37Q8C4×C4⋊C4C428C4C429C4C23.65C23C23.83C23C42C2×C4C22C22
# reps12414441611

Matrix representation of C42.37Q8 in GL6(𝔽5)

320000
020000
000400
001000
000001
000040
,
100000
010000
004000
000400
000001
000040
,
300000
120000
000100
001000
000020
000002
,
410000
010000
002000
000200
000010
000004

G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;

C42.37Q8 in GAP, Magma, Sage, TeX

C_4^2._{37}Q_8
% in TeX

G:=Group("C4^2.37Q8");
// GroupNames label

G:=SmallGroup(128,1303);
// by ID

G=gap.SmallGroup(128,1303);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,268,675,80]);
// Polycyclic

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

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