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G = C2×C42.2C22order 128 = 27

Direct product of C2 and C42.2C22

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

Aliases: C2×C42.2C22, C42.65D4, C42.141C23, C22.39C4≀C2, C42.82(C2×C4), C42.C2.9C4, (C22×C4).664D4, C8⋊C4.141C22, (C2×C42).185C22, C42.C2.91C22, C23.220(C22⋊C4), C22.18(C4.10D4), C2.28(C2×C4≀C2), (C2×C4⋊C4).16C4, C4⋊C4.22(C2×C4), (C2×C8⋊C4).19C2, (C2×C4).1169(C2×D4), C2.9(C2×C4.10D4), (C2×C42.C2).2C2, (C2×C4).135(C22×C4), (C22×C4).207(C2×C4), (C2×C4).176(C22⋊C4), C22.199(C2×C22⋊C4), SmallGroup(128,255)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C42.2C22
C1C2C22C2×C4C42C2×C42C2×C42.C2 — C2×C42.2C22
C1C22C2×C4 — C2×C42.2C22
C1C23C2×C42 — C2×C42.2C22
C1C22C22C42 — C2×C42.2C22

Generators and relations for C2×C42.2C22
 G = < a,b,c,d,e | a2=b4=c4=1, d2=c, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe-1=b-1, cd=dc, ece-1=b2c-1, ede-1=b-1c2d >

Subgroups: 196 in 116 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2 [×6], C4 [×10], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], C23, C42 [×2], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×12], C22×C4, C22×C4 [×2], C22×C4 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C42.2C22 [×4], C2×C8⋊C4 [×2], C2×C42.C2, C2×C42.2C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C42.2C22 [×4], C2×C4.10D4, C2×C4≀C2 [×2], C2×C42.2C22

Smallest permutation representation of C2×C42.2C22
Regular action on 128 points
Generators in S128
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 117)(10 118)(11 119)(12 120)(13 113)(14 114)(15 115)(16 116)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 109)(26 110)(27 111)(28 112)(29 105)(30 106)(31 107)(32 108)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(49 125)(50 126)(51 127)(52 128)(53 121)(54 122)(55 123)(56 124)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(73 84)(74 85)(75 86)(76 87)(77 88)(78 81)(79 82)(80 83)
(1 39 111 95)(2 36 112 92)(3 33 105 89)(4 38 106 94)(5 35 107 91)(6 40 108 96)(7 37 109 93)(8 34 110 90)(9 17 121 88)(10 22 122 85)(11 19 123 82)(12 24 124 87)(13 21 125 84)(14 18 126 81)(15 23 127 86)(16 20 128 83)(25 57 97 72)(26 62 98 69)(27 59 99 66)(28 64 100 71)(29 61 101 68)(30 58 102 65)(31 63 103 70)(32 60 104 67)(41 49 73 113)(42 54 74 118)(43 51 75 115)(44 56 76 120)(45 53 77 117)(46 50 78 114)(47 55 79 119)(48 52 80 116)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(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)
(1 119 5 115)(2 76 6 80)(3 53 7 49)(4 42 8 46)(9 25 13 29)(10 69 14 65)(11 103 15 99)(12 60 16 64)(17 72 21 68)(18 102 22 98)(19 63 23 59)(20 28 24 32)(26 81 30 85)(27 123 31 127)(33 45 37 41)(34 114 38 118)(35 75 39 79)(36 56 40 52)(43 95 47 91)(44 108 48 112)(50 94 54 90)(51 111 55 107)(57 84 61 88)(58 122 62 126)(66 82 70 86)(67 128 71 124)(73 89 77 93)(74 110 78 106)(83 100 87 104)(92 120 96 116)(97 125 101 121)(105 117 109 113)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111112224
type++++++-
imageC1C2C2C2C4C4D4D4C4≀C2C4.10D4
kernelC2×C42.2C22C42.2C22C2×C8⋊C4C2×C42.C2C2×C4⋊C4C42.C2C42C22×C4C22C22
# reps14214422162

Matrix representation of C2×C42.2C22 in GL6(𝔽17)

1600000
0160000
0016000
0001600
000010
000001
,
400000
0130000
0016000
0001600
0000169
0000131
,
1300000
0130000
0016000
0001600
0000132
000014
,
0130000
100000
0001600
001000
000003
00001012
,
0160000
100000
001000
0001600
00001110
000056

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,1,0,0,0,0,2,4],[0,1,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,10,0,0,0,0,3,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,11,5,0,0,0,0,10,6] >;

C2×C42.2C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._2C_2^2
% in TeX

G:=Group("C2xC4^2.2C2^2");
// GroupNames label

G:=SmallGroup(128,255);
// by ID

G=gap.SmallGroup(128,255);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971,102]);
// Polycyclic

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

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