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G = C4.6Q32order 128 = 27

2nd non-split extension by C4 of Q32 acting via Q32/Q16=C2

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

Aliases: C4.6Q32, C8.14Q16, C4.7SD32, C8.25SD16, C42.40D4, C4⋊C16.6C2, C81C8.4C2, C2.D8.5C4, (C2×C8).336D4, (C2×C4).122D8, C82Q8.2C2, (C4×C8).36C22, (C2×C4).20SD16, C4.3(Q8⋊C4), C2.5(C2.Q32), C2.5(M5(2)⋊C2), C2.5(C4.10D8), C4.3(C4.10D4), C22.64(D4⋊C4), (C2×C8).26(C2×C4), (C2×C4).226(C22⋊C4), SmallGroup(128,97)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4.6Q32
C1C2C4C2×C4C2×C8C4×C8C82Q8 — C4.6Q32
C1C2C2×C4C2×C8 — C4.6Q32
C1C22C42C4×C8 — C4.6Q32
C1C2C2C2C2C2×C4C2×C4C4×C8 — C4.6Q32

Generators and relations for C4.6Q32
 G = < a,b,c | a4=b16=1, c2=a2b8, bab-1=cac-1=a-1, cbc-1=a-1b-1 >

2C4
8C4
8C4
2C8
4C2×C4
4C2×C4
8Q8
8C8
8Q8
2C4⋊C4
2C4⋊C4
4C4⋊C4
4C2×C8
4C2×Q8
4C16
2C4⋊C8
2C4⋊Q8
2C2.D8
2C2×C16

Character table of C4.6Q32

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111222241616222244888844444444
ρ111111111111111111111111111111    trivial
ρ2111111111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1111111-1-1-1-111111111    linear of order 2
ρ51111-1-111-11-1-1-1-1-111ii-i-ii-i-iii-ii-i    linear of order 4
ρ61111-1-111-1-11-1-1-1-111ii-i-i-iii-i-ii-ii    linear of order 4
ρ71111-1-111-1-11-1-1-1-111-i-iiii-i-iii-ii-i    linear of order 4
ρ81111-1-111-11-1-1-1-1-111-i-iii-iii-i-ii-ii    linear of order 4
ρ92222-2-222-2002222-2-2000000000000    orthogonal lifted from D4
ρ1022222222200-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ11222222-2-2-20000000000002-22-222-2-2    orthogonal lifted from D8
ρ12222222-2-2-2000000000000-22-22-2-222    orthogonal lifted from D8
ρ1322-2-2-22000002-22-22-20000ζ1615169ζ1651631615169ζ1651631615169ζ1615169165163165163    symplectic lifted from Q32, Schur index 2
ρ142-2-22002-200022-2-2002-22-200000000    symplectic lifted from Q16, Schur index 2
ρ1522-2-2-2200000-22-22-220000165163ζ1615169ζ165163ζ1615169ζ16516316516316151691615169    symplectic lifted from Q32, Schur index 2
ρ1622-2-2-2200000-22-22-220000ζ16516316151691651631615169165163ζ165163ζ1615169ζ1615169    symplectic lifted from Q32, Schur index 2
ρ172-2-22002-200022-2-200-22-2200000000    symplectic lifted from Q16, Schur index 2
ρ1822-2-2-22000002-22-22-200001615169165163ζ1615169165163ζ16151691615169ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ192222-2-2-2-22000000000000--2--2-2-2--2-2-2--2    complex lifted from SD16
ρ202-2-22002-2000-2-22200--2-2-2--200000000    complex lifted from SD16
ρ212-2-22002-2000-2-22200-2--2--2-200000000    complex lifted from SD16
ρ222222-2-2-2-22000000000000-2-2--2--2-2--2--2-2    complex lifted from SD16
ρ2322-2-22-200000-22-222-20000ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ165163ζ1615169ζ16716    complex lifted from SD32
ρ2422-2-22-2000002-22-2-220000ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ16716ζ165163ζ16131611    complex lifted from SD32
ρ2522-2-22-200000-22-222-20000ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ16131611ζ16716ζ1615169    complex lifted from SD32
ρ2622-2-22-2000002-22-2-220000ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ1615169ζ16131611ζ165163    complex lifted from SD32
ρ274-44-40000000-222222-2200000000000000    orthogonal lifted from M5(2)⋊C2
ρ284-44-4000000022-22-222200000000000000    orthogonal lifted from M5(2)⋊C2
ρ294-4-4400-44000000000000000000000    symplectic lifted from C4.10D4, Schur index 2

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

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

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

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

Matrix representation of C4.6Q32 in GL4(𝔽17) generated by

16000
01600
00161
00151
,
71600
1700
00130
0094
,
11000
101600
0005
00100
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,15,0,0,1,1],[7,1,0,0,16,7,0,0,0,0,13,9,0,0,0,4],[1,10,0,0,10,16,0,0,0,0,0,10,0,0,5,0] >;

C4.6Q32 in GAP, Magma, Sage, TeX

C_4._6Q_{32}
% in TeX

G:=Group("C4.6Q32");
// GroupNames label

G:=SmallGroup(128,97);
// by ID

G=gap.SmallGroup(128,97);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,794,416,2028,124]);
// Polycyclic

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

Export

Subgroup lattice of C4.6Q32 in TeX
Character table of C4.6Q32 in TeX

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