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G = C42.7Q8order 128 = 27

7th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Aliases: C42.7Q8, C42.28D4, C8⋊C410C4, (C2×C4).7C42, C22.33C4≀C2, C42.35(C2×C4), C428C4.2C2, (C22×C4).639D4, C2.9(C426C4), (C2×C42).132C22, C2.6(C22.C42), C23.215(C22⋊C4), C22.24(C4.D4), C22.15(C4.10D4), C2.2(C42.2C22), C2.3(C42.C22), C22.45(C2.C42), (C2×C4⋊C4).4C4, (C2×C4).20(C4⋊C4), (C2×C8⋊C4).15C2, (C22×C4).150(C2×C4), (C2×C4).300(C22⋊C4), SmallGroup(128,27)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.7Q8
C1C2C22C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.7Q8
C1C22C2×C4 — C42.7Q8
C1C23C2×C42 — C42.7Q8
C1C22C22C2×C42 — C42.7Q8

Generators and relations for C42.7Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=ac3 >

Subgroups: 168 in 90 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×8], C22 [×3], C22 [×4], C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], C23, C42 [×2], C42 [×2], C4⋊C4 [×2], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×2], C2.C42 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×2], C22×C8 [×2], C428C4, C2×C8⋊C4 [×2], C42.7Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C4≀C2 [×4], C42.C22 [×2], C42.2C22 [×2], C426C4 [×2], C22.C42, C42.7Q8

Smallest permutation representation of C42.7Q8
Regular action on 128 points
Generators in S128
(1 57 51 41)(2 62 52 46)(3 59 53 43)(4 64 54 48)(5 61 55 45)(6 58 56 42)(7 63 49 47)(8 60 50 44)(9 24 82 26)(10 21 83 31)(11 18 84 28)(12 23 85 25)(13 20 86 30)(14 17 87 27)(15 22 88 32)(16 19 81 29)(33 68 90 73)(34 65 91 78)(35 70 92 75)(36 67 93 80)(37 72 94 77)(38 69 95 74)(39 66 96 79)(40 71 89 76)(97 118 124 105)(98 115 125 110)(99 120 126 107)(100 117 127 112)(101 114 128 109)(102 119 121 106)(103 116 122 111)(104 113 123 108)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 125 37 121)(34 126 38 122)(35 127 39 123)(36 128 40 124)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)(65 107 69 111)(66 108 70 112)(67 109 71 105)(68 110 72 106)(73 115 77 119)(74 116 78 120)(75 117 79 113)(76 118 80 114)(89 97 93 101)(90 98 94 102)(91 99 95 103)(92 100 96 104)
(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 33 17 123)(2 71 18 111)(3 92 19 98)(4 78 20 118)(5 37 21 127)(6 67 22 107)(7 96 23 102)(8 74 24 114)(9 97 44 91)(10 113 45 73)(11 126 46 36)(12 110 47 70)(13 101 48 95)(14 117 41 77)(15 122 42 40)(16 106 43 66)(25 121 49 39)(26 109 50 69)(27 104 51 90)(28 116 52 76)(29 125 53 35)(30 105 54 65)(31 100 55 94)(32 120 56 80)(34 82 124 60)(38 86 128 64)(57 72 87 112)(58 89 88 103)(59 79 81 119)(61 68 83 108)(62 93 84 99)(63 75 85 115)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I4J4K4L4M4N8A···8P
order12···24···44444448···8
size11···12···24488884···4

38 irreducible representations

dim11111222244
type++++-++-
imageC1C2C2C4C4D4Q8D4C4≀C2C4.D4C4.10D4
kernelC42.7Q8C428C4C2×C8⋊C4C8⋊C4C2×C4⋊C4C42C42C22×C4C22C22C22
# reps112841121611

Matrix representation of C42.7Q8 in GL6(𝔽17)

1600000
0160000
0016000
0013100
0000130
0000013
,
1600000
0160000
0013000
0001300
0000160
0000101
,
100000
13160000
0013200
007400
0000130
000081
,
420000
0130000
0013200
001400
0000102
0000107

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,10,0,0,0,0,0,1],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,13,7,0,0,0,0,2,4,0,0,0,0,0,0,13,8,0,0,0,0,0,1],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,13,1,0,0,0,0,2,4,0,0,0,0,0,0,10,10,0,0,0,0,2,7] >;

C42.7Q8 in GAP, Magma, Sage, TeX

C_4^2._7Q_8
% in TeX

G:=Group("C4^2.7Q8");
// GroupNames label

G:=SmallGroup(128,27);
// by ID

G=gap.SmallGroup(128,27);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924,172]);
// Polycyclic

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

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