Copied to
clipboard

G = C4.6Q32  order 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, C8⋊1C8.4C2, C2.D8.5C4, (C2×C8).336D4, (C2×C4).122D8, C8⋊2Q8.2C2, (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

Derived series
C1 — C2×C8 — C4.6Q32
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8⋊2Q8 — C4.6Q32
Lower central
C1 — C2 — C2×C4 — C2×C8 — C4.6Q32
Upper central
C1 — C22 — C42 — C4×C8 — C4.6Q32
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×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 >

C1
C2
C2
C2
C4
C22
C4
C4
C4
2C4
8C4
8C4
C8
C8
C2×C4
C2×C4
C2×C4
2C8
4C2×C4
4C2×C4
8Q8
8C8
8Q8
C2×C8
C42
C2×C8
2C4⋊C4
2C4⋊C4
4C4⋊C4
4C2×C8
4C2×Q8
4C16
C4×C8
C2.D8
C2.D8
2C4⋊C8
2C4⋊Q8
2C2.D8
2C2×C16
C4⋊C16
C8⋊1C8
C8⋊2Q8
C4.6Q32

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
ρ11222222-2-2-2000000000000√2-√2√2-√2√2√2-√2-√2    orthogonal lifted from D8
ρ12222222-2-2-2000000000000-√2√2-√2√2-√2-√2√2√2    orthogonal lifted from D8
ρ1322-2-2-2200000√2-√2√2-√2√2-√20000ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169ζ1615-ζ169-ζ165+ζ163-ζ165+ζ163    symplectic lifted from Q32, Schur index 2
ρ142-2-22002-200022-2-200√2-√2√2-√200000000    symplectic lifted from Q16, Schur index 2
ρ1522-2-2-2200000-√2√2-√2√2-√2√20000-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163-ζ165+ζ163-ζ1615+ζ169-ζ1615+ζ169    symplectic lifted from Q32, Schur index 2
ρ1622-2-2-2200000-√2√2-√2√2-√2√20000ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163ζ165-ζ163ζ1615-ζ169ζ1615-ζ169    symplectic lifted from Q32, Schur index 2
ρ172-2-22002-200022-2-200-√2√2-√2√200000000    symplectic lifted from Q16, Schur index 2
ρ1822-2-2-2200000√2-√2√2-√2√2-√20000-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169-ζ1615+ζ169ζ165-ζ163ζ165-ζ163    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-√2√2-√2√2√2-√20000ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ1615+ζ169ζ167+ζ16    complex lifted from SD32
ρ2422-2-22-200000√2-√2√2-√2-√2√20000ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ165+ζ163ζ1613+ζ1611    complex lifted from SD32
ρ2522-2-22-200000-√2√2-√2√2√2-√20000ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169    complex lifted from SD32
ρ2622-2-22-200000√2-√2√2-√2-√2√20000ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163    complex lifted from SD32
ρ274-44-40000000-2√22√22√2-2√200000000000000    orthogonal lifted from M5(2)⋊C2
ρ284-44-400000002√2-2√2-2√22√200000000000000    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
(1 101 69 28)(2 29 70 102)(3 103 71 30)(4 31 72 104)(5 105 73 32)(6 17 74 106)(7 107 75 18)(8 19 76 108)(9 109 77 20)(10 21 78 110)(11 111 79 22)(12 23 80 112)(13 97 65 24)(14 25 66 98)(15 99 67 26)(16 27 68 100)(33 86 60 127)(34 128 61 87)(35 88 62 113)(36 114 63 89)(37 90 64 115)(38 116 49 91)(39 92 50 117)(40 118 51 93)(41 94 52 119)(42 120 53 95)(43 96 54 121)(44 122 55 81)(45 82 56 123)(46 124 57 83)(47 84 58 125)(48 126 59 85)

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

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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁