Copied to
clipboard

G = C427Q8order 128 = 27

7th semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C427Q8, C23.451C24, C22.2362+ 1+4, C22.1822- 1+4, C2.5Q82, C2.30D42, C4⋊C419Q8, C42(C4⋊Q8), C2.24(D4×Q8), C4⋊C4.234D4, C429C4.28C2, (C22×C4).840C23, (C2×C42).556C22, C22.302(C22×D4), C22.101(C22×Q8), (C22×Q8).134C22, C23.78C23.7C2, C23.65C23.53C2, C2.C42.188C22, C2.12(C23.41C23), (C4×C4⋊C4).64C2, C2.14(C2×C4⋊Q8), (C2×C4).77(C2×D4), (C2×C4⋊Q8).33C2, (C2×C4).51(C2×Q8), (C2×C4⋊C4).871C22, SmallGroup(128,1283)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C427Q8
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C427Q8
C1C23 — C427Q8
C1C23 — C427Q8
C1C23 — C427Q8

Generators and relations for C427Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 468 in 278 conjugacy classes, 140 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×30], C2×C4 [×30], Q8 [×16], C23, C42 [×4], C42 [×8], C4⋊C4 [×16], C4⋊C4 [×26], C22×C4, C22×C4 [×14], C2×Q8 [×20], C2.C42 [×8], C2×C42, C2×C42 [×4], C2×C4⋊C4 [×18], C4⋊Q8 [×16], C22×Q8 [×4], C4×C4⋊C4 [×2], C429C4, C23.65C23 [×4], C23.78C23 [×4], C2×C4⋊Q8 [×4], C427Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×12], C23 [×15], C2×D4 [×12], C2×Q8 [×18], C24, C4⋊Q8 [×8], C22×D4 [×2], C22×Q8 [×3], 2+ 1+4, 2- 1+4, C2×C4⋊Q8 [×2], C23.41C23, D42, D4×Q8 [×2], Q82, C427Q8

Smallest permutation representation of C427Q8
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 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 37 127 36)(6 38 128 33)(7 39 125 34)(8 40 126 35)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)(61 65 100 72)(62 66 97 69)(63 67 98 70)(64 68 99 71)(73 108 111 86)(74 105 112 87)(75 106 109 88)(76 107 110 85)(77 113 82 102)(78 114 83 103)(79 115 84 104)(80 116 81 101)(89 117 96 124)(90 118 93 121)(91 119 94 122)(92 120 95 123)
(1 98 58 52)(2 62 59 43)(3 100 60 50)(4 64 57 41)(5 101 109 122)(6 115 110 118)(7 103 111 124)(8 113 112 120)(9 18 72 28)(10 14 69 32)(11 20 70 26)(12 16 71 30)(13 67 31 21)(15 65 29 23)(17 66 27 24)(19 68 25 22)(33 79 107 90)(34 83 108 96)(35 77 105 92)(36 81 106 94)(37 80 88 91)(38 84 85 93)(39 78 86 89)(40 82 87 95)(42 47 61 53)(44 45 63 55)(46 97 56 51)(48 99 54 49)(73 117 125 114)(74 123 126 102)(75 119 127 116)(76 121 128 104)
(1 119 58 116)(2 123 59 102)(3 117 60 114)(4 121 57 104)(5 63 109 44)(6 99 110 49)(7 61 111 42)(8 97 112 51)(9 39 72 86)(10 35 69 105)(11 37 70 88)(12 33 71 107)(13 94 31 81)(14 92 32 77)(15 96 29 83)(16 90 30 79)(17 95 27 82)(18 89 28 78)(19 93 25 84)(20 91 26 80)(21 36 67 106)(22 38 68 85)(23 34 65 108)(24 40 66 87)(41 128 64 76)(43 126 62 74)(45 122 55 101)(46 120 56 113)(47 124 53 103)(48 118 54 115)(50 125 100 73)(52 127 98 75)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111122244
type++++++-+-+-
imageC1C2C2C2C2C2Q8D4Q82+ 1+42- 1+4
kernelC427Q8C4×C4⋊C4C429C4C23.65C23C23.78C23C2×C4⋊Q8C42C4⋊C4C4⋊C4C22C22
# reps12144448811

Matrix representation of C427Q8 in GL6(𝔽5)

420000
410000
004400
002100
000002
000020
,
130000
140000
004400
002100
000010
000001
,
130000
140000
001000
000100
000001
000040
,
340000
020000
004000
002100
000003
000030

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

C427Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7Q_8
% in TeX

G:=Group("C4^2:7Q8");
// GroupNames label

G:=SmallGroup(128,1283);
// by ID

G=gap.SmallGroup(128,1283);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675,80]);
// 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*b^2,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