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G = C42.40Q8order 128 = 27

40th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.40Q8, C23.745C24, C22.5182+ 1+4, C22.3972- 1+4, C428C4.52C2, (C2×C42).748C22, (C22×C4).256C23, C22.177(C22×Q8), C2.C42.446C22, C23.81C23.54C2, C2.8(C22.58C24), C2.64(C22.56C24), C2.52(C23.41C23), (C2×C4).140(C2×Q8), (C2×C4⋊C4).552C22, SmallGroup(128,1577)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.40Q8
C1C2C22C23C22×C4C2×C4⋊C4C23.81C23 — C42.40Q8
C1C23 — C42.40Q8
C1C23 — C42.40Q8
C1C23 — C42.40Q8

Generators and relations for C42.40Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 308 in 170 conjugacy classes, 92 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×18], C22, C22 [×6], C2×C4 [×6], C2×C4 [×42], C23, C42 [×4], C4⋊C4 [×18], C22×C4 [×15], C2.C42 [×16], C2×C42, C2×C4⋊C4 [×18], C428C4 [×3], C23.81C23 [×12], C42.40Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, 2+ 1+4 [×3], 2- 1+4 [×3], C23.41C23 [×3], C22.56C24 [×3], C22.58C24, C42.40Q8

Character table of C42.40Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-111-1-11111-1-1-1-111-1-1    linear of order 2
ρ31111111111-1-1-1-1-111-11-11-1-11-11    linear of order 2
ρ411111111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ511111111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ611111111-1-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ71111111111-1-1-1-11-11-1-11-111-1-11    linear of order 2
ρ811111111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111111-1-111-1-111-1-1-1-111-1-111    linear of order 2
ρ111111111111-1-1-1-1-11-111-1-111-11-1    linear of order 2
ρ1211111111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ1311111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1411111111-1-111-1-1-1-1-1-111-1-11111    linear of order 2
ρ151111111111-1-1-1-11-1-11-111-1-111-1    linear of order 2
ρ1611111111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ172-22-22-22-2-222-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-22-22-22-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-22-22-2-22-222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-22-22-22-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2444-444-4-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ25444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

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

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

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

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

Matrix representation of C42.40Q8 in GL12(𝔽5)

011300000000
403400000000
420100000000
214000000000
000040400000
000010320000
000020100000
000002400000
000000000142
000000002442
000000000133
000000001123
,
010000000000
400000000000
000400000000
001000000000
000012000000
000044000000
000033010000
000002400000
000000000100
000000001000
000000004440
000000000441
,
001000000000
000100000000
400000000000
040000000000
000043000000
000001000000
000022040000
000033400000
000000000010
000000004440
000000001000
000000001041
,
310100000000
124000000000
043100000000
101200000000
000030300000
000000110000
000000200000
000004300000
000000000001
000000000114
000000001041
000000001000

G:=sub<GL(12,GF(5))| [0,4,4,2,0,0,0,0,0,0,0,0,1,0,2,1,0,0,0,0,0,0,0,0,1,3,0,4,0,0,0,0,0,0,0,0,3,4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,3,1,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,4,1,1,0,0,0,0,0,0,0,0,4,4,3,2,0,0,0,0,0,0,0,0,2,2,3,3],[0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,3,0,0,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,0,4,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,2,3,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,4,0,4,0,0,0,0,0,0,0,0,0,0,0,1],[3,1,0,1,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,0,0,4,3,1,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,4,1,0] >;

C42.40Q8 in GAP, Magma, Sage, TeX

C_4^2._{40}Q_8
% in TeX

G:=Group("C4^2.40Q8");
// GroupNames label

G:=SmallGroup(128,1577);
// by ID

G=gap.SmallGroup(128,1577);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,456,758,723,436,794,185,80]);
// Polycyclic

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

Export

Character table of C42.40Q8 in TeX

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