Copied to
clipboard

G = C4.(C4⋊Q8)  order 128 = 27

8th non-split extension by C4 of C4⋊Q8 acting via C4⋊Q8/C2×Q8=C2

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

Aliases: C4⋊C4.4Q8, (C2×C8).166D4, C4.37(C4⋊Q8), (C2×C4).42SD16, C2.23(C88D4), C2.23(C8⋊D4), C2.9(Q8⋊Q8), C2.9(D42Q8), (C22×C4).323D4, C23.938(C2×D4), C2.13(D4.Q8), C2.13(Q8.Q8), C4.24(C42.C2), C22.128(C4○D8), C22.4Q16.29C2, (C22×C8).328C22, (C2×C42).389C22, C22.107(C2×SD16), C22.261(C4⋊D4), C22.157(C8⋊C22), (C22×C4).1472C23, C22.115(C22⋊Q8), C4.117(C22.D4), C22.146(C8.C22), C23.65C23.25C2, C2.10(C23.81C23), (C2×C4⋊C8).49C2, (C2×C4).289(C2×Q8), (C2×C4.Q8).27C2, (C2×C4).1381(C2×D4), (C2×C4).631(C4○D4), (C2×C4⋊C4).159C22, SmallGroup(128,820)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4.(C4⋊Q8)
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.65C23 — C4.(C4⋊Q8)
C1C2C22×C4 — C4.(C4⋊Q8)
C1C23C2×C42 — C4.(C4⋊Q8)
C1C2C2C22×C4 — C4.(C4⋊Q8)

Generators and relations for C4.(C4⋊Q8)
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a-1c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a-1b-1, dcd-1=c3 >

Subgroups: 224 in 112 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×19], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×3], C23.65C23 [×2], C2×C4⋊C8, C2×C4.Q8, C4.(C4⋊Q8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, SD16 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.81C23, C88D4, C8⋊D4, Q8⋊Q8, D42Q8, D4.Q8, Q8.Q8, C4.(C4⋊Q8)

Smallest permutation representation of C4.(C4⋊Q8)
Regular action on 128 points
Generators in S128
(1 81 5 85)(2 82 6 86)(3 83 7 87)(4 84 8 88)(9 89 13 93)(10 90 14 94)(11 91 15 95)(12 92 16 96)(17 74 21 78)(18 75 22 79)(19 76 23 80)(20 77 24 73)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(41 99 45 103)(42 100 46 104)(43 101 47 97)(44 102 48 98)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 115 61 119)(58 116 62 120)(59 117 63 113)(60 118 64 114)(65 126 69 122)(66 127 70 123)(67 128 71 124)(68 121 72 125)
(1 122 47 17)(2 66 48 75)(3 128 41 23)(4 72 42 73)(5 126 43 21)(6 70 44 79)(7 124 45 19)(8 68 46 77)(9 38 63 110)(10 25 64 49)(11 36 57 108)(12 31 58 55)(13 34 59 106)(14 29 60 53)(15 40 61 112)(16 27 62 51)(18 82 123 98)(20 88 125 104)(22 86 127 102)(24 84 121 100)(26 115 50 91)(28 113 52 89)(30 119 54 95)(32 117 56 93)(33 120 105 96)(35 118 107 94)(37 116 109 92)(39 114 111 90)(65 101 74 85)(67 99 76 83)(69 97 78 81)(71 103 80 87)
(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 35 87 27)(2 38 88 30)(3 33 81 25)(4 36 82 28)(5 39 83 31)(6 34 84 26)(7 37 85 29)(8 40 86 32)(9 68 95 127)(10 71 96 122)(11 66 89 125)(12 69 90 128)(13 72 91 123)(14 67 92 126)(15 70 93 121)(16 65 94 124)(17 64 80 120)(18 59 73 115)(19 62 74 118)(20 57 75 113)(21 60 76 116)(22 63 77 119)(23 58 78 114)(24 61 79 117)(41 105 97 49)(42 108 98 52)(43 111 99 55)(44 106 100 50)(45 109 101 53)(46 112 102 56)(47 107 103 51)(48 110 104 54)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim1111122222244
type+++++-+++-
imageC1C2C2C2C2Q8D4D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernelC4.(C4⋊Q8)C22.4Q16C23.65C23C2×C4⋊C8C2×C4.Q8C4⋊C4C2×C8C22×C4C2×C4C2×C4C22C22C22
# reps1321142246411

Matrix representation of C4.(C4⋊Q8) in GL6(𝔽17)

100000
010000
001000
000100
0000130
000004
,
1600000
0160000
00121200
0012500
000002
000080
,
400000
16130000
0016000
0001600
000080
000002
,
1320000
040000
0016200
000100
000001
000010

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

C4.(C4⋊Q8) in GAP, Magma, Sage, TeX

C_4.(C_4\rtimes Q_8)
% in TeX

G:=Group("C4.(C4:Q8)");
// GroupNames label

G:=SmallGroup(128,820);
// by ID

G=gap.SmallGroup(128,820);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^-1*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽