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G = C42.23Q8order 128 = 27

23rd non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.23Q8, C42.103D4, C424C4.7C2, C2.5(C428C4), C4.81(C4.4D4), C22.48(C8○D4), (C22×C8).20C22, C4.35(C42.C2), C2.C42.17C4, C23.309(C22×C4), (C2×C42).254C22, (C22×C4).1622C23, C22.7C42.4C2, C22.80(C42⋊C2), C2.8(C42.7C22), C2.11(C42.6C22), (C2×C4⋊C8).25C2, (C2×C4).44(C4⋊C4), C22.95(C2×C4⋊C4), (C2×C4).338(C2×Q8), (C2×C4).1520(C2×D4), (C2×C4).930(C4○D4), (C22×C4).114(C2×C4), SmallGroup(128,564)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.23Q8
C1C2C4C2×C4C22×C4C2×C42C424C4 — C42.23Q8
C1C23 — C42.23Q8
C1C22×C4 — C42.23Q8
C1C2C2C22×C4 — C42.23Q8

Generators and relations for C42.23Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 172 in 108 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×6], C2.C42 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C22×C8 [×4], C22.7C42 [×4], C424C4, C2×C4⋊C8 [×2], C42.23Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C8○D4 [×4], C428C4, C42.6C22 [×2], C42.7C22 [×4], C42.23Q8

Smallest permutation representation of C42.23Q8
Regular action on 128 points
Generators in S128
(1 65 47 14)(2 15 48 66)(3 67 41 16)(4 9 42 68)(5 69 43 10)(6 11 44 70)(7 71 45 12)(8 13 46 72)(17 125 103 75)(18 76 104 126)(19 127 97 77)(20 78 98 128)(21 121 99 79)(22 80 100 122)(23 123 101 73)(24 74 102 124)(25 61 111 95)(26 96 112 62)(27 63 105 89)(28 90 106 64)(29 57 107 91)(30 92 108 58)(31 59 109 93)(32 94 110 60)(33 83 119 49)(34 50 120 84)(35 85 113 51)(36 52 114 86)(37 87 115 53)(38 54 116 88)(39 81 117 55)(40 56 118 82)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 122 13 126)(10 123 14 127)(11 124 15 128)(12 125 16 121)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)(65 77 69 73)(66 78 70 74)(67 79 71 75)(68 80 72 76)(81 105 85 109)(82 106 86 110)(83 107 87 111)(84 108 88 112)(89 113 93 117)(90 114 94 118)(91 115 95 119)(92 116 96 120)
(1 119 103 89)(2 64 104 34)(3 113 97 91)(4 58 98 36)(5 115 99 93)(6 60 100 38)(7 117 101 95)(8 62 102 40)(9 26 128 56)(10 83 121 105)(11 28 122 50)(12 85 123 107)(13 30 124 52)(14 87 125 109)(15 32 126 54)(16 81 127 111)(17 63 47 33)(18 120 48 90)(19 57 41 35)(20 114 42 92)(21 59 43 37)(22 116 44 94)(23 61 45 39)(24 118 46 96)(25 67 55 77)(27 69 49 79)(29 71 51 73)(31 65 53 75)(66 110 76 88)(68 112 78 82)(70 106 80 84)(72 108 74 86)
(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)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4H4I···4T8A···8P
order12···24···44···48···8
size11···11···14···44···4

44 irreducible representations

dim111112222
type+++++-
imageC1C2C2C2C4D4Q8C4○D4C8○D4
kernelC42.23Q8C22.7C42C424C4C2×C4⋊C8C2.C42C42C42C2×C4C22
# reps1412822816

Matrix representation of C42.23Q8 in GL6(𝔽17)

040000
400000
0013000
009400
0000111
0000616
,
1600000
0160000
0016000
0001600
000040
000004
,
010000
100000
0013000
0001300
0000713
0000410
,
530000
14120000
0041300
0081300
000038
0000914

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,13,9,0,0,0,0,0,4,0,0,0,0,0,0,1,6,0,0,0,0,11,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,7,4,0,0,0,0,13,10],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,4,8,0,0,0,0,13,13,0,0,0,0,0,0,3,9,0,0,0,0,8,14] >;

C42.23Q8 in GAP, Magma, Sage, TeX

C_4^2._{23}Q_8
% in TeX

G:=Group("C4^2.23Q8");
// GroupNames label

G:=SmallGroup(128,564);
// by ID

G=gap.SmallGroup(128,564);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,124]);
// Polycyclic

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

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