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G = C4212Q8order 128 = 27

12nd semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C4212Q8, C23.743C24, C22.5162+ 1+4, C22.3952- 1+4, C425C4.19C2, C429C4.40C2, (C22×C4).254C23, (C2×C42).746C22, C22.175(C22×Q8), (C22×Q8).245C22, C2.59(C22.54C24), C23.83C23.50C2, C23.81C23.53C2, C2.C42.444C22, C23.78C23.31C2, C2.75(C22.57C24), C2.50(C23.41C23), (C2×C4).138(C2×Q8), (C2×C4⋊C4).550C22, SmallGroup(128,1575)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4212Q8
C1C2C22C23C22×C4C2×C4⋊C4C23.78C23 — C4212Q8
C1C23 — C4212Q8
C1C23 — C4212Q8
C1C23 — C4212Q8

Generators and relations for C4212Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 324 in 174 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×18], C22, C22 [×6], C2×C4 [×6], C2×C4 [×42], Q8 [×4], C23, C42 [×4], C4⋊C4 [×15], C22×C4, C22×C4 [×14], C2×Q8 [×3], C2.C42 [×18], C2×C42, C2×C4⋊C4 [×15], C22×Q8, C425C4 [×2], C429C4, C23.78C23 [×3], C23.81C23 [×6], C23.83C23 [×3], C4212Q8
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], C22.54C24, C22.57C24 [×3], C4212Q8

Character table of C4212Q8

 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 C4212Q8
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 75 38 10)(2 76 39 11)(3 73 40 12)(4 74 37 9)(5 41 70 102)(6 42 71 103)(7 43 72 104)(8 44 69 101)(13 110 78 49)(14 111 79 50)(15 112 80 51)(16 109 77 52)(17 45 82 106)(18 46 83 107)(19 47 84 108)(20 48 81 105)(21 120 86 59)(22 117 87 60)(23 118 88 57)(24 119 85 58)(25 55 90 116)(26 56 91 113)(27 53 92 114)(28 54 89 115)(29 68 94 128)(30 65 95 125)(31 66 96 126)(32 67 93 127)(33 124 98 63)(34 121 99 64)(35 122 100 61)(36 123 97 62)
(1 14 6 107)(2 13 7 106)(3 16 8 105)(4 15 5 108)(9 112 102 19)(10 111 103 18)(11 110 104 17)(12 109 101 20)(21 31 114 124)(22 30 115 123)(23 29 116 122)(24 32 113 121)(25 35 118 128)(26 34 119 127)(27 33 120 126)(28 36 117 125)(37 80 70 47)(38 79 71 46)(39 78 72 45)(40 77 69 48)(41 84 74 51)(42 83 75 50)(43 82 76 49)(44 81 73 52)(53 63 86 96)(54 62 87 95)(55 61 88 94)(56 64 85 93)(57 68 90 100)(58 67 91 99)(59 66 92 98)(60 65 89 97)
(1 22 6 115)(2 88 7 55)(3 24 8 113)(4 86 5 53)(9 57 102 90)(10 119 103 26)(11 59 104 92)(12 117 101 28)(13 61 106 94)(14 123 107 30)(15 63 108 96)(16 121 105 32)(17 66 110 98)(18 127 111 34)(19 68 112 100)(20 125 109 36)(21 70 114 37)(23 72 116 39)(25 74 118 41)(27 76 120 43)(29 78 122 45)(31 80 124 47)(33 82 126 49)(35 84 128 51)(38 87 71 54)(40 85 69 56)(42 91 75 58)(44 89 73 60)(46 95 79 62)(48 93 77 64)(50 99 83 67)(52 97 81 65)

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

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

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

Matrix representation of C4212Q8 in GL10(𝔽5)

4000000000
0400000000
0040240000
0013240000
0042320000
0004400000
0000002202
0000000003
0000000233
0000000300
,
4000000000
0400000000
0043000000
0011000000
0044040000
0001100000
0000001200
0000000400
0000000012
0000000004
,
0100000000
4000000000
0010300000
0000110000
0000400000
0001100000
0000000010
0000000001
0000004000
0000000400
,
2000000000
0300000000
0023230000
0021020000
0000100000
0022310000
0000001441
0000004411
0000004141
0000001111

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

C4212Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{12}Q_8
% in TeX

G:=Group("C4^2:12Q8");
// GroupNames label

G:=SmallGroup(128,1575);
// by ID

G=gap.SmallGroup(128,1575);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,352,794,185,80]);
// 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,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of C4212Q8 in TeX

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