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G = C42.9C23order 128 = 27

9th non-split extension by C42 of C23 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.9C23, C8⋊Q8.1C2, C4⋊C4.44D4, C4⋊Q8.39C22, C8⋊C4.2C22, C42.2C22.C2, C2.24(D4.8D4), C22.190C22≀C2, C42.C2.7C22, C22.58C24.C2, C2.22(D4.10D4), C42.30C22.4C2, (C2×C4).222(C2×D4), SmallGroup(128,395)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.9C23
C1C2C22C2×C4C42C42.C2C22.58C24 — C42.9C23
C1C22C42 — C42.9C23
C1C22C42 — C42.9C23
C1C22C22C42 — C42.9C23

Generators and relations for C42.9C23
 G = < a,b,c,d,e | a4=b4=1, c2=e2=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, eae-1=a-1b2, cbc-1=ebe-1=b-1, dbd-1=a2b-1, dcd-1=ac, ece-1=bc, de=ed >

Subgroups: 168 in 86 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C2 [×2], C4 [×10], C22, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×7], Q8 [×2], C42, C42 [×2], C4⋊C4 [×6], C4⋊C4 [×14], C2×C8 [×3], C2×Q8, C8⋊C4, C8⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C42.C2, C42.C2 [×2], C42.C2 [×6], C4⋊Q8, C42.2C22, C42.2C22 [×2], C42.30C22, C8⋊Q8 [×2], C22.58C24, C42.9C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D4.8D4, D4.10D4 [×2], C42.9C23

Character table of C42.9C23

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F
 size 111144488888816888888
ρ111111111111111111111    trivial
ρ211111111-1-11-1-111-1-1-1-11    linear of order 2
ρ311111111-1-11-1-1-1-11111-1    linear of order 2
ρ41111111111111-1-1-1-1-1-1-1    linear of order 2
ρ51111111-11-1-11-1-11-11-111    linear of order 2
ρ61111111-1-11-1-11-111-11-11    linear of order 2
ρ71111111-1-11-1-111-1-11-11-1    linear of order 2
ρ81111111-11-1-11-11-11-11-1-1    linear of order 2
ρ92222-2-2200200-20000000    orthogonal lifted from D4
ρ1022222-2-2-2002000000000    orthogonal lifted from D4
ρ1122222-2-2200-2000000000    orthogonal lifted from D4
ρ122222-2-2200-20020000000    orthogonal lifted from D4
ρ132222-22-20-200200000000    orthogonal lifted from D4
ρ142222-22-20200-200000000    orthogonal lifted from D4
ρ154-44-4000000000020000-2    symplectic lifted from D4.10D4, Schur index 2
ρ164-44-40000000000-200002    symplectic lifted from D4.10D4, Schur index 2
ρ1744-4-40000000000020-200    symplectic lifted from D4.10D4, Schur index 2
ρ1844-4-400000000000-20200    symplectic lifted from D4.10D4, Schur index 2
ρ194-4-440000000000002i0-2i0    complex lifted from D4.8D4
ρ204-4-44000000000000-2i02i0    complex lifted from D4.8D4

Smallest permutation representation of C42.9C23
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 20 13 5)(2 17 14 6)(3 18 15 7)(4 19 16 8)(9 121 118 128)(10 122 119 125)(11 123 120 126)(12 124 117 127)(21 29 35 26)(22 30 36 27)(23 31 33 28)(24 32 34 25)(37 42 47 50)(38 43 48 51)(39 44 45 52)(40 41 46 49)(53 58 67 63)(54 59 68 64)(55 60 65 61)(56 57 66 62)(69 76 79 84)(70 73 80 81)(71 74 77 82)(72 75 78 83)(85 92 97 93)(86 89 98 94)(87 90 99 95)(88 91 100 96)(101 116 111 106)(102 113 112 107)(103 114 109 108)(104 115 110 105)
(1 71 3 69)(2 70 4 72)(5 74 7 76)(6 73 8 75)(9 68 11 66)(10 67 12 65)(13 77 15 79)(14 80 16 78)(17 81 19 83)(18 84 20 82)(21 85 23 87)(22 88 24 86)(25 89 27 91)(26 92 28 90)(29 93 31 95)(30 96 32 94)(33 99 35 97)(34 98 36 100)(37 101 39 103)(38 104 40 102)(41 107 43 105)(42 106 44 108)(45 109 47 111)(46 112 48 110)(49 113 51 115)(50 116 52 114)(53 117 55 119)(54 120 56 118)(57 121 59 123)(58 124 60 122)(61 125 63 127)(62 128 64 126)
(1 45 15 37)(2 48 16 40)(3 47 13 39)(4 46 14 38)(5 50 18 44)(6 49 19 43)(7 52 20 42)(8 51 17 41)(9 97 120 87)(10 100 117 86)(11 99 118 85)(12 98 119 88)(21 67 33 55)(22 66 34 54)(23 65 35 53)(24 68 36 56)(25 62 30 59)(26 61 31 58)(27 64 32 57)(28 63 29 60)(69 110 77 102)(70 109 78 101)(71 112 79 104)(72 111 80 103)(73 116 83 108)(74 115 84 107)(75 114 81 106)(76 113 82 105)(89 127 96 122)(90 126 93 121)(91 125 94 124)(92 128 95 123)
(1 29 3 31)(2 25 4 27)(5 35 7 33)(6 24 8 22)(9 102 11 104)(10 111 12 109)(13 26 15 28)(14 32 16 30)(17 34 19 36)(18 23 20 21)(37 63 39 61)(38 57 40 59)(41 54 43 56)(42 67 44 65)(45 60 47 58)(46 64 48 62)(49 68 51 66)(50 53 52 55)(69 99 71 97)(70 86 72 88)(73 94 75 96)(74 92 76 90)(77 85 79 87)(78 100 80 98)(81 89 83 91)(82 93 84 95)(101 117 103 119)(105 121 107 123)(106 127 108 125)(110 118 112 120)(113 126 115 128)(114 122 116 124)

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

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

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

Matrix representation of C42.9C23 in GL8(𝔽17)

101500000
001610000
101600000
1161600000
00000010
000041640
00001000
0000111001
,
115000000
116000000
016010000
1161600000
00000100
000016000
000041640
00001201213
,
981120000
2103150000
71515100000
4131300000
00008028
000015111515
000051288
00001014137
,
793120000
151516130000
67820000
1331440000
000028615
00008409
0000166015
000068611
,
912230000
8148110000
6161040000
1651210000
00000001
00005054
000067016
00001000

G:=sub<GL(8,GF(17))| [1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,1,11,0,0,0,0,0,16,0,10,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,4,12,0,0,0,0,1,0,16,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,0,13],[9,2,7,4,0,0,0,0,8,10,15,13,0,0,0,0,1,3,15,13,0,0,0,0,12,15,10,0,0,0,0,0,0,0,0,0,8,15,5,10,0,0,0,0,0,11,12,14,0,0,0,0,2,15,8,13,0,0,0,0,8,15,8,7],[7,15,6,13,0,0,0,0,9,15,7,3,0,0,0,0,3,16,8,14,0,0,0,0,12,13,2,4,0,0,0,0,0,0,0,0,2,8,16,6,0,0,0,0,8,4,6,8,0,0,0,0,6,0,0,6,0,0,0,0,15,9,15,11],[9,8,6,16,0,0,0,0,12,14,16,5,0,0,0,0,2,8,10,12,0,0,0,0,3,11,4,1,0,0,0,0,0,0,0,0,0,5,6,1,0,0,0,0,0,0,7,0,0,0,0,0,0,5,0,0,0,0,0,0,1,4,16,0] >;

C42.9C23 in GAP, Magma, Sage, TeX

C_4^2._9C_2^3
% in TeX

G:=Group("C4^2.9C2^3");
// GroupNames label

G:=SmallGroup(128,395);
// by ID

G=gap.SmallGroup(128,395);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,184,1123,570,521,136,3924,1411,998,242]);
// Polycyclic

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

Export

Character table of C42.9C23 in TeX

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