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G = C42.515C23order 128 = 27

376th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.515C23, C4.362- 1+4, C4⋊C4.180D4, C84Q8.5C2, Q8.Q8.2C2, Q8⋊Q8.1C2, (C4×Q16).20C2, Q83Q8.5C2, (C2×Q8).246D4, C2.64(Q8○D8), C4⋊C4.440C23, C4⋊C8.139C22, (C2×C8).120C23, (C4×C8).233C22, (C2×C4).566C24, Q8.38(C4○D4), Q16⋊C4.1C2, C42Q16.12C2, C4⋊Q8.195C22, C8⋊C4.65C22, C4.Q8.75C22, C2.74(Q85D4), C4.80(C8.C22), C4.SD16.10C2, (C2×Q8).260C23, (C4×Q8).197C22, (C2×Q16).92C22, C2.D8.207C22, Q8⋊C4.91C22, C22.826(C22×D4), C42.C2.68C22, C42.30C22.1C2, C4.267(C2×C4○D4), (C2×C4).642(C2×D4), C2.88(C2×C8.C22), SmallGroup(128,2106)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.515C23
C1C2C4C2×C4C42C4×Q8Q83Q8 — C42.515C23
C1C2C2×C4 — C42.515C23
C1C22C4×Q8 — C42.515C23
C1C2C2C2×C4 — C42.515C23

Generators and relations for C42.515C23
 G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >

Subgroups: 256 in 163 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×14], C22, C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×2], Q8 [×11], C42, C42 [×2], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×17], C2×C8 [×2], C2×C8 [×2], Q16 [×6], C2×Q8 [×3], C2×Q8 [×2], C2×Q8, C4×C8, C8⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×8], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×2], C2.D8, C4×Q8 [×3], C4×Q8 [×4], C4×Q8 [×2], C42.C2 [×2], C42.C2 [×5], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×Q16, C2×Q16 [×2], C4×Q16, Q16⋊C4 [×2], C84Q8, C42Q16, C42Q16 [×2], Q8⋊Q8, Q8.Q8 [×2], C4.SD16, C42.30C22 [×2], Q83Q8 [×2], C42.515C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C2×C8.C22, Q8○D8, C42.515C23

Character table of C42.515C23

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D8E8F
 size 11112222444444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-111-1-111-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ31111-11-11-11-11-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ411111111-1-111-1-11111-1-1-1-111111-1-1    linear of order 2
ρ511111111-1-11-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ61111-11-11-11-1-1-111-11-111-1-11-1-1111-1    linear of order 2
ρ71111-11-111-1-1-11-11-11-1-1-1111-1-111-11    linear of order 2
ρ811111111111-111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ911111111-1-111-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ101111-11-11-11-11-11111-1-11-11-111-1-1-11    linear of order 2
ρ111111-11-111-1-111-1111-11-11-1-111-1-11-1    linear of order 2
ρ1211111111111111-11-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-11-111-1-1-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ151111-11-11-11-1-1-11-1-1-111-11-1111-1-1-11    linear of order 2
ρ1611111111-1-11-1-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ1722222-22-222-20-2-2000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-22-220-22000000000000000    orthogonal lifted from D4
ρ1922222-22-2-2-2-2022000000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-22202-2000000000000000    orthogonal lifted from D4
ρ212-22-2020-2000-200-2i22i0000002i-2i0000    complex lifted from C4○D4
ρ222-22-2020-20002002i-2-2i0000002i-2i0000    complex lifted from C4○D4
ρ232-22-2020-2000-2002i2-2i000000-2i2i0000    complex lifted from C4○D4
ρ242-22-2020-2000200-2i-22i000000-2i2i0000    complex lifted from C4○D4
ρ254-4-4440-40000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-40-404000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ274-4-44-4040000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C42.515C23
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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 128 9 15)(6 125 10 16)(7 126 11 13)(8 127 12 14)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)(61 73 71 65)(62 74 72 66)(63 75 69 67)(64 76 70 68)(77 81 89 87)(78 82 90 88)(79 83 91 85)(80 84 92 86)(93 99 105 101)(94 100 106 102)(95 97 107 103)(96 98 108 104)(109 121 115 117)(110 122 116 118)(111 123 113 119)(112 124 114 120)
(1 89 25 77)(2 90 26 78)(3 91 27 79)(4 92 28 80)(5 76 9 68)(6 73 10 65)(7 74 11 66)(8 75 12 67)(13 72 126 62)(14 69 127 63)(15 70 128 64)(16 71 125 61)(17 87 23 81)(18 88 24 82)(19 85 21 83)(20 86 22 84)(29 105 41 93)(30 106 42 94)(31 107 43 95)(32 108 44 96)(33 103 39 97)(34 104 40 98)(35 101 37 99)(36 102 38 100)(45 115 51 109)(46 116 52 110)(47 113 49 111)(48 114 50 112)(53 121 57 117)(54 122 58 118)(55 123 59 119)(56 124 60 120)
(1 53 3 55)(2 56 4 54)(5 106 7 108)(6 105 8 107)(9 94 11 96)(10 93 12 95)(13 98 15 100)(14 97 16 99)(17 45 19 47)(18 48 20 46)(21 49 23 51)(22 52 24 50)(25 57 27 59)(26 60 28 58)(29 65 31 67)(30 68 32 66)(33 69 35 71)(34 72 36 70)(37 61 39 63)(38 64 40 62)(41 73 43 75)(42 76 44 74)(77 119 79 117)(78 118 80 120)(81 111 83 109)(82 110 84 112)(85 115 87 113)(86 114 88 116)(89 123 91 121)(90 122 92 124)(101 127 103 125)(102 126 104 128)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 114 9 112)(6 115 10 109)(7 116 11 110)(8 113 12 111)(13 118 126 122)(14 119 127 123)(15 120 128 124)(16 117 125 121)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)(45 61 51 71)(46 62 52 72)(47 63 49 69)(48 64 50 70)(53 65 57 73)(54 66 58 74)(55 67 59 75)(56 68 60 76)(77 99 89 101)(78 100 90 102)(79 97 91 103)(80 98 92 104)(81 93 87 105)(82 94 88 106)(83 95 85 107)(84 96 86 108)

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

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

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

Matrix representation of C42.515C23 in GL6(𝔽17)

16160000
210000
0016000
0001600
0000160
0000016
,
100000
010000
000100
0016000
0000016
000010
,
13130000
840000
0013598
005499
00991213
0089135
,
0120000
700000
00301216
0003165
001216140
00165014
,
100000
010000
000010
000001
0016000
0001600

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

C42.515C23 in GAP, Magma, Sage, TeX

C_4^2._{515}C_2^3
% in TeX

G:=Group("C4^2.515C2^3");
// GroupNames label

G:=SmallGroup(128,2106);
// by ID

G=gap.SmallGroup(128,2106);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,352,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.515C23 in TeX

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