Copied to
clipboard

G = C426Q8order 128 = 27

6th semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C426Q8, C23.450C24, C22.2352+ 1+4, C22.1812- 1+4, C4⋊C418Q8, C4⋊C4.233D4, C4.26(C4⋊Q8), C2.72(D45D4), C428C4.32C2, C2.16(Q83Q8), C2.31(D43Q8), C2.39(Q85D4), (C22×C4).839C23, (C2×C42).555C22, C22.301(C22×D4), C22.100(C22×Q8), (C22×Q8).133C22, C23.78C23.6C2, C23.67C23.39C2, C2.C42.187C22, C23.65C23.52C2, C23.81C23.14C2, C2.26(C23.37C23), C2.11(C23.41C23), (C4×C4⋊C4).63C2, C2.13(C2×C4⋊Q8), (C2×C4).76(C2×D4), (C2×C4⋊Q8).32C2, (C2×C4).125(C2×Q8), (C2×C4).149(C4○D4), (C2×C4⋊C4).304C22, C22.327(C2×C4○D4), (C2×C42.C2).19C2, SmallGroup(128,1282)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C426Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C426Q8
C1C23 — C426Q8
C1C23 — C426Q8
C1C23 — C426Q8

Generators and relations for C426Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 388 in 234 conjugacy classes, 120 normal (42 characteristic)
C1, C2 [×7], C4 [×4], C4 [×22], C22 [×7], C2×C4 [×22], C2×C4 [×34], Q8 [×8], C23, C42 [×4], C42 [×6], C4⋊C4 [×8], C4⋊C4 [×26], C22×C4 [×7], C22×C4 [×8], C2×Q8 [×10], C2.C42 [×2], C2.C42 [×10], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×6], C2×C4⋊C4 [×10], C42.C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C4×C4⋊C4 [×2], C428C4, C23.65C23 [×2], C23.65C23 [×2], C23.67C23 [×2], C23.78C23 [×2], C23.81C23 [×2], C2×C42.C2, C2×C4⋊Q8, C426Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C2×C4⋊Q8, C23.37C23, C23.41C23, D45D4, Q85D4, D43Q8, Q83Q8, C426Q8

Smallest permutation representation of C426Q8
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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 59 51 45)(2 58 52 48)(3 57 49 47)(4 60 50 46)(5 42 62 56)(6 41 63 55)(7 44 64 54)(8 43 61 53)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(13 71 27 39)(14 70 28 38)(15 69 25 37)(16 72 26 40)(33 105 98 95)(34 108 99 94)(35 107 100 93)(36 106 97 96)(65 117 125 75)(66 120 126 74)(67 119 127 73)(68 118 128 76)(77 85 115 123)(78 88 116 122)(79 87 113 121)(80 86 114 124)(81 89 103 111)(82 92 104 110)(83 91 101 109)(84 90 102 112)
(1 83 51 101)(2 116 52 78)(3 81 49 103)(4 114 50 80)(5 126 62 66)(6 33 63 98)(7 128 64 68)(8 35 61 100)(9 115 23 77)(10 84 24 102)(11 113 21 79)(12 82 22 104)(13 119 27 73)(14 96 28 106)(15 117 25 75)(16 94 26 108)(17 123 31 85)(18 92 32 110)(19 121 29 87)(20 90 30 112)(34 72 99 40)(36 70 97 38)(37 125 69 65)(39 127 71 67)(41 95 55 105)(42 120 56 74)(43 93 53 107)(44 118 54 76)(45 91 59 109)(46 124 60 86)(47 89 57 111)(48 122 58 88)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim111111111222244
type+++++++++-+-+-
imageC1C2C2C2C2C2C2C2C2Q8D4Q8C4○D42+ 1+42- 1+4
kernelC426Q8C4×C4⋊C4C428C4C23.65C23C23.67C23C23.78C23C23.81C23C2×C42.C2C2×C4⋊Q8C42C4⋊C4C4⋊C4C2×C4C22C22
# reps121422211444811

Matrix representation of C426Q8 in GL6(𝔽5)

430000
110000
004000
001100
000010
000001
,
100000
010000
002000
003300
000010
000001
,
100000
440000
003000
002200
000030
000032
,
400000
040000
004300
001100
000042
000041

G:=sub<GL(6,GF(5))| [4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,4,0,0,0,0,2,1] >;

C426Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6Q_8
% in TeX

G:=Group("C4^2:6Q8");
// GroupNames label

G:=SmallGroup(128,1282);
// by ID

G=gap.SmallGroup(128,1282);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,675,80]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽