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G = C4213Q8order 128 = 27

13rd semidirect product of C42 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4213Q8, C23.744C24, C22.3962- 1+4, C22.5172+ 1+4, C428C4.51C2, (C22×C4).255C23, (C2×C42).747C22, C22.176(C22×Q8), (C22×Q8).246C22, C2.22(C24⋊C22), C2.C42.445C22, C23.78C23.32C2, C23.83C23.51C2, C2.76(C22.57C24), C2.51(C23.41C23), (C2×C4).139(C2×Q8), (C2×C4⋊C4).551C22, SmallGroup(128,1576)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4213Q8
C1C2C22C23C22×C4C2×C4⋊C4C23.78C23 — C4213Q8
C1C23 — C4213Q8
C1C23 — C4213Q8
C1C23 — C4213Q8

Generators and relations for C4213Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 340 in 178 conjugacy classes, 92 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×18], C22, C22 [×6], C2×C4 [×6], C2×C4 [×42], Q8 [×8], C23, C42 [×4], C4⋊C4 [×12], C22×C4 [×15], C2×Q8 [×6], C2.C42 [×20], C2×C42, C2×C4⋊C4 [×12], C22×Q8 [×2], C428C4 [×3], C23.78C23 [×6], C23.83C23 [×6], C4213Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, 2+ 1+4 [×3], 2- 1+4 [×3], C23.41C23 [×3], C24⋊C22, C22.57C24 [×3], C4213Q8

Character table of C4213Q8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-111-1-11111-1-1-1-111-1-1    linear of order 2
ρ31111111111-1-1-1-1-111-11-11-1-11-11    linear of order 2
ρ411111111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ511111111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ611111111-1-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ71111111111-1-1-1-11-11-1-11-111-1-11    linear of order 2
ρ811111111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111111-1-111-1-111-1-1-1-111-1-111    linear of order 2
ρ111111111111-1-1-1-1-11-111-1-111-11-1    linear of order 2
ρ1211111111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ1311111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1411111111-1-111-1-1-1-1-1-111-1-11111    linear of order 2
ρ151111111111-1-1-1-11-1-11-111-1-111-1    linear of order 2
ρ1611111111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ172-22-22-22-2-222-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-22-22-22-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-22-22-2-22-222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-22-22-22-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-444-4-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C4213Q8
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 76 103 13)(2 73 104 14)(3 74 101 15)(4 75 102 16)(5 36 70 95)(6 33 71 96)(7 34 72 93)(8 35 69 94)(9 107 40 48)(10 108 37 45)(11 105 38 46)(12 106 39 47)(17 56 80 115)(18 53 77 116)(19 54 78 113)(20 55 79 114)(21 111 84 52)(22 112 81 49)(23 109 82 50)(24 110 83 51)(25 121 88 62)(26 122 85 63)(27 123 86 64)(28 124 87 61)(29 58 92 117)(30 59 89 118)(31 60 90 119)(32 57 91 120)(41 126 97 68)(42 127 98 65)(43 128 99 66)(44 125 100 67)
(1 19 11 52)(2 77 12 110)(3 17 9 50)(4 79 10 112)(5 30 99 63)(6 92 100 121)(7 32 97 61)(8 90 98 123)(13 54 46 21)(14 116 47 83)(15 56 48 23)(16 114 45 81)(18 39 51 104)(20 37 49 102)(22 75 55 108)(24 73 53 106)(25 33 58 67)(26 95 59 128)(27 35 60 65)(28 93 57 126)(29 44 62 71)(31 42 64 69)(34 120 68 87)(36 118 66 85)(38 111 103 78)(40 109 101 80)(41 124 72 91)(43 122 70 89)(74 115 107 82)(76 113 105 84)(86 94 119 127)(88 96 117 125)
(1 27 11 60)(2 87 12 120)(3 25 9 58)(4 85 10 118)(5 53 99 24)(6 113 100 84)(7 55 97 22)(8 115 98 82)(13 62 46 29)(14 122 47 89)(15 64 48 31)(16 124 45 91)(17 67 50 33)(18 126 51 93)(19 65 52 35)(20 128 49 95)(21 71 54 44)(23 69 56 42)(26 37 59 102)(28 39 57 104)(30 73 63 106)(32 75 61 108)(34 77 68 110)(36 79 66 112)(38 119 103 86)(40 117 101 88)(41 81 72 114)(43 83 70 116)(74 123 107 90)(76 121 105 92)(78 127 111 94)(80 125 109 96)

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

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

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

Matrix representation of C4213Q8 in GL10(𝔽5)

4000000000
0400000000
0000130000
0024210000
0003020000
0024010000
0000000132
0000003440
0000002001
0000000331
,
1000000000
0100000000
0013000000
0014000000
0032430000
0010110000
0000004200
0000000100
0000004110
0000002414
,
2000000000
2300000000
0040400000
0000010000
0000100000
0001000000
0000004030
0000001330
0000001010
0000000412
,
3400000000
0200000000
0042030000
0001100000
0000400000
0000110000
0000004144
0000002134
0000001422
0000003123

G:=sub<GL(10,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,4,3,4,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,3,1,2,1,0,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,4,0,3,0,0,0,0,0,0,3,4,0,3,0,0,0,0,0,0,2,0,1,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,3,1,0,0,0,0,0,0,3,4,2,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,4,0,4,2,0,0,0,0,0,0,2,1,1,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,4],[2,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,3,3,1,1,0,0,0,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,0,4,2,1,3,0,0,0,0,0,0,1,1,4,1,0,0,0,0,0,0,4,3,2,2,0,0,0,0,0,0,4,4,2,3] >;

C4213Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{13}Q_8
% in TeX

G:=Group("C4^2:13Q8");
// GroupNames label

G:=SmallGroup(128,1576);
// by ID

G=gap.SmallGroup(128,1576);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,456,758,723,352,794,185,80]);
// Polycyclic

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

Export

Character table of C4213Q8 in TeX

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