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G = C4.Q32order 128 = 27

3rd non-split extension by C4 of Q32 acting via Q32/Q16=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q162Q8, C4.7Q32, C42.148D4, (C2×C8).72D4, C4⋊C16.10C2, C2.7(C2×Q32), C8.32(C2×Q8), (C2×C4).154D8, (C4×Q16).7C2, C163C4.4C2, C8.66(C4○D4), (C2×C16).8C22, (C4×C8).68C22, C82Q8.12C2, (C2×C8).529C23, C2.Q32.3C2, C22.115(C2×D8), C4.48(C22⋊Q8), C2.15(C16⋊C22), C2.D8.14C22, C2.15(D4⋊Q8), C4.11(C8.C22), (C2×Q16).111C22, (C2×C4).797(C2×D4), SmallGroup(128,959)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4.Q32
C1C2C4C8C2×C8C2×Q16C4×Q16 — C4.Q32
C1C2C4C2×C8 — C4.Q32
C1C22C42C4×C8 — C4.Q32
C1C2C2C2C2C4C4C2×C8 — C4.Q32

Generators and relations for C4.Q32
 G = < a,b,c | a4=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 148 in 65 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×5], C16 [×2], C42, C42, C4⋊C4 [×5], C2×C8 [×2], Q16 [×2], Q16, C2×Q8 [×2], C4×C8, Q8⋊C4, C2.D8, C2.D8 [×2], C2.D8, C2×C16 [×2], C4×Q8, C4⋊Q8, C2×Q16, C2.Q32 [×2], C4⋊C16, C163C4 [×2], C4×Q16, C82Q8, C4.Q32
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D8 [×2], C2×D4, C2×Q8, C4○D4, Q32 [×2], C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C2×Q32, C16⋊C22, C4.Q32

Character table of C4.Q32

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1-11-1-1-111-111111-1-1-1111-1-1-11    linear of order 2
ρ411111-1-11-1-1-1111-11111-1-11-1-1-1111-1    linear of order 2
ρ511111-1-11-111-1-1-111111-1-11-1-1-1111-1    linear of order 2
ρ611111-1-11-111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ7111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ92222-2-2-2-220000000000002-22-2-22-22    orthogonal lifted from D8
ρ102222-222-2-200000000000022-22-22-2-2    orthogonal lifted from D8
ρ11222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1222222-2-22-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ132222-222-2-2000000000000-2-22-22-222    orthogonal lifted from D8
ρ142222-2-2-2-22000000000000-22-222-22-2    orthogonal lifted from D8
ρ152-2-22200-202-200002-22-20000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22200-20-2200002-22-20000000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-20-2200000000-2-222-22ζ165163ζ16516316716165163ζ1671616516316716ζ16716    symplectic lifted from Q32, Schur index 2
ρ1822-2-20-2200000000-2-222-22165163165163ζ16716ζ16516316716ζ165163ζ1671616716    symplectic lifted from Q32, Schur index 2
ρ1922-2-202-20000000022-2-2-2216716ζ16716ζ16516316716ζ165163ζ16716165163165163    symplectic lifted from Q32, Schur index 2
ρ2022-2-20-220000000022-2-22-21671616716165163ζ16716ζ165163ζ16716165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ2122-2-202-200000000-2-2222-2165163ζ1651631671616516316716ζ165163ζ16716ζ16716    symplectic lifted from Q32, Schur index 2
ρ2222-2-202-200000000-2-2222-2ζ165163165163ζ16716ζ165163ζ167161651631671616716    symplectic lifted from Q32, Schur index 2
ρ2322-2-20-220000000022-2-22-2ζ16716ζ16716ζ1651631671616516316716ζ165163165163    symplectic lifted from Q32, Schur index 2
ρ2422-2-202-20000000022-2-2-22ζ1671616716165163ζ1671616516316716ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ252-2-22200-2000-2i2i00-22-220000000000    complex lifted from C4○D4
ρ262-2-22200-20002i-2i00-22-220000000000    complex lifted from C4○D4
ρ274-44-40000000000022-22-22220000000000    orthogonal lifted from C16⋊C22
ρ284-44-400000000000-222222-220000000000    orthogonal lifted from C16⋊C22
ρ294-4-44-4004000000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C4.Q32
Regular action on 128 points
Generators in S128
(1 59 70 22)(2 23 71 60)(3 61 72 24)(4 25 73 62)(5 63 74 26)(6 27 75 64)(7 49 76 28)(8 29 77 50)(9 51 78 30)(10 31 79 52)(11 53 80 32)(12 17 65 54)(13 55 66 18)(14 19 67 56)(15 57 68 20)(16 21 69 58)(33 84 102 115)(34 116 103 85)(35 86 104 117)(36 118 105 87)(37 88 106 119)(38 120 107 89)(39 90 108 121)(40 122 109 91)(41 92 110 123)(42 124 111 93)(43 94 112 125)(44 126 97 95)(45 96 98 127)(46 128 99 81)(47 82 100 113)(48 114 101 83)
(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 45 9 37)(2 97 10 105)(3 43 11 35)(4 111 12 103)(5 41 13 33)(6 109 14 101)(7 39 15 47)(8 107 16 99)(17 85 25 93)(18 115 26 123)(19 83 27 91)(20 113 28 121)(21 81 29 89)(22 127 30 119)(23 95 31 87)(24 125 32 117)(34 73 42 65)(36 71 44 79)(38 69 46 77)(40 67 48 75)(49 90 57 82)(50 120 58 128)(51 88 59 96)(52 118 60 126)(53 86 61 94)(54 116 62 124)(55 84 63 92)(56 114 64 122)(66 102 74 110)(68 100 76 108)(70 98 78 106)(72 112 80 104)

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

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

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

Matrix representation of C4.Q32 in GL4(𝔽17) generated by

16000
01600
0069
001111
,
11400
131100
00153
00162
,
121200
12500
0072
001010
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,6,11,0,0,9,11],[11,13,0,0,4,11,0,0,0,0,15,16,0,0,3,2],[12,12,0,0,12,5,0,0,0,0,7,10,0,0,2,10] >;

C4.Q32 in GAP, Magma, Sage, TeX

C_4.Q_{32}
% in TeX

G:=Group("C4.Q32");
// GroupNames label

G:=SmallGroup(128,959);
// by ID

G=gap.SmallGroup(128,959);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,504,141,64,422,352,1684,438,242,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations

Export

Character table of C4.Q32 in TeX

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