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G = C2×C2.Q32order 128 = 27

Direct product of C2 and C2.Q32

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×C2.Q32, C23.58D8, C22.5Q32, C22.11SD32, (C2×Q16)⋊9C4, Q167(C2×C4), (C2×C4).78D8, C8.91(C2×D4), C2.1(C2×Q32), (C2×C8).247D4, C2.2(C2×SD32), C4.9(C2×SD16), (C22×C16).7C2, C8.28(C22×C4), (C2×C4).76SD16, C22.52(C2×D8), C8.25(C22⋊C4), (C2×C8).493C23, (C2×C16).62C22, (C22×C4).583D4, (C22×Q16).6C2, C4.24(D4⋊C4), (C2×Q16).98C22, C2.D8.142C22, (C22×C8).527C22, C22.53(D4⋊C4), (C2×C8).175(C2×C4), (C2×C4).755(C2×D4), C4.49(C2×C22⋊C4), (C2×C2.D8).22C2, C2.27(C2×D4⋊C4), (C2×C4).270(C22⋊C4), SmallGroup(128,869)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C2×C2.Q32
C1C2C4C2×C4C2×C8C22×C8C22×Q16 — C2×C2.Q32
C1C2C4C8 — C2×C2.Q32
C1C23C22×C4C22×C8 — C2×C2.Q32
C1C2C2C2C2C4C4C2×C8 — C2×C2.Q32

Generators and relations for C2×C2.Q32
 G = < a,b,c,d | a2=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 244 in 116 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C16, C4⋊C4, C2×C8, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×Q8, C2.D8, C2.D8, C2×C16, C2×C16, C2×C4⋊C4, C22×C8, C2×Q16, C2×Q16, C22×Q8, C2.Q32, C2×C2.D8, C22×C16, C22×Q16, C2×C2.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, SD32, Q32, C2×C22⋊C4, C2×D8, C2×SD16, C2.Q32, C2×D4⋊C4, C2×SD32, C2×Q32, C2×C2.Q32

Smallest permutation representation of C2×C2.Q32
Regular action on 128 points
Generators in S128
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 65)(61 66)(62 67)(63 68)(64 69)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 113)
(1 93)(2 94)(3 95)(4 96)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 124)(50 125)(51 126)(52 127)(53 128)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 97)(77 98)(78 99)(79 100)(80 101)
(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 76 9 68)(2 112 10 104)(3 74 11 66)(4 110 12 102)(5 72 13 80)(6 108 14 100)(7 70 15 78)(8 106 16 98)(17 116 25 124)(18 56 26 64)(19 114 27 122)(20 54 28 62)(21 128 29 120)(22 52 30 60)(23 126 31 118)(24 50 32 58)(33 49 41 57)(34 123 42 115)(35 63 43 55)(36 121 44 113)(37 61 45 53)(38 119 46 127)(39 59 47 51)(40 117 48 125)(65 96 73 88)(67 94 75 86)(69 92 77 84)(71 90 79 82)(81 109 89 101)(83 107 91 99)(85 105 93 97)(87 103 95 111)

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

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

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

44 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L8A···8H16A···16P
order12···244444···48···816···16
size11···122228···82···22···2

44 irreducible representations

dim1111112222222
type+++++++++-
imageC1C2C2C2C2C4D4D4D8SD16D8SD32Q32
kernelC2×C2.Q32C2.Q32C2×C2.D8C22×C16C22×Q16C2×Q16C2×C8C22×C4C2×C4C2×C4C23C22C22
# reps1411183124288

Matrix representation of C2×C2.Q32 in GL4(𝔽17) generated by

1000
01600
00160
00016
,
16000
0100
0010
0001
,
4000
0100
001311
00613
,
16000
01600
00512
001212
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,13,6,0,0,11,13],[16,0,0,0,0,16,0,0,0,0,5,12,0,0,12,12] >;

C2×C2.Q32 in GAP, Magma, Sage, TeX

C_2\times C_2.Q_{32}
% in TeX

G:=Group("C2xC2.Q32");
// GroupNames label

G:=SmallGroup(128,869);
// by ID

G=gap.SmallGroup(128,869);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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