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G = C4×Q32order 128 = 27

Direct product of C4 and Q32

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

Aliases: C4×Q32, C42.331D4, (C4×C16).8C2, C4.27(C4×D4), C2.15(C4×D8), C2.3(C2×Q32), C16.13(C2×C4), C43(C163C4), (C2×C8).233D4, (C2×C4).174D8, (C2×Q32).7C2, (C4×Q16).4C2, Q16.2(C2×C4), C163C4.9C2, C8.40(C4○D4), C4.13(C4○D8), C2.5(C4○D16), C8.37(C22×C4), C22.63(C2×D8), C43(C2.Q32), (C4×C8).398C22, (C2×C8).504C23, (C2×C16).71C22, C2.Q32.8C2, C2.D8.152C22, (C2×Q16).103C22, (C2×C4).770(C2×D4), SmallGroup(128,906)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4×Q32
C1C2C4C2×C4C2×C8C4×C8C4×Q16 — C4×Q32
C1C2C4C8 — C4×Q32
C1C2×C4C42C4×C8 — C4×Q32
C1C2C2C2C2C4C4C2×C8 — C4×Q32

Generators and relations for C4×Q32
 G = < a,b,c | a4=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 148 in 75 conjugacy classes, 42 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×7], C22, C8 [×2], C8, C2×C4 [×3], C2×C4 [×4], Q8 [×6], C16 [×2], C16, C42, C42 [×2], C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], Q16 [×2], C2×Q8 [×2], C4×C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], Q32 [×4], C4×Q8 [×2], C2×Q16 [×2], C4×C16, C2.Q32 [×2], C163C4, C4×Q16 [×2], C2×Q32, C4×Q32
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, Q32 [×2], C4×D4, C2×D8, C4○D8, C4×D8, C2×Q32, C4○D16, C4×Q32

Smallest permutation representation of C4×Q32
Regular action on 128 points
Generators in S128
(1 24 77 101)(2 25 78 102)(3 26 79 103)(4 27 80 104)(5 28 65 105)(6 29 66 106)(7 30 67 107)(8 31 68 108)(9 32 69 109)(10 17 70 110)(11 18 71 111)(12 19 72 112)(13 20 73 97)(14 21 74 98)(15 22 75 99)(16 23 76 100)(33 117 93 50)(34 118 94 51)(35 119 95 52)(36 120 96 53)(37 121 81 54)(38 122 82 55)(39 123 83 56)(40 124 84 57)(41 125 85 58)(42 126 86 59)(43 127 87 60)(44 128 88 61)(45 113 89 62)(46 114 90 63)(47 115 91 64)(48 116 92 49)
(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 84 9 92)(2 83 10 91)(3 82 11 90)(4 81 12 89)(5 96 13 88)(6 95 14 87)(7 94 15 86)(8 93 16 85)(17 64 25 56)(18 63 26 55)(19 62 27 54)(20 61 28 53)(21 60 29 52)(22 59 30 51)(23 58 31 50)(24 57 32 49)(33 76 41 68)(34 75 42 67)(35 74 43 66)(36 73 44 65)(37 72 45 80)(38 71 46 79)(39 70 47 78)(40 69 48 77)(97 128 105 120)(98 127 106 119)(99 126 107 118)(100 125 108 117)(101 124 109 116)(102 123 110 115)(103 122 111 114)(104 121 112 113)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P8A···8H16A···16P
order1222444444444···48···816···16
size1111111122228···82···22···2

44 irreducible representations

dim11111112222222
type+++++++++-
imageC1C2C2C2C2C2C4D4D4C4○D4D8Q32C4○D8C4○D16
kernelC4×Q32C4×C16C2.Q32C163C4C4×Q16C2×Q32Q32C42C2×C8C8C2×C4C4C4C2
# reps11212181124848

Matrix representation of C4×Q32 in GL4(𝔽17) generated by

4000
0400
00160
00016
,
14300
141400
00613
0046
,
14300
3300
00125
0055
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,6,4,0,0,13,6],[14,3,0,0,3,3,0,0,0,0,12,5,0,0,5,5] >;

C4×Q32 in GAP, Magma, Sage, TeX

C_4\times Q_{32}
% in TeX

G:=Group("C4xQ32");
// GroupNames label

G:=SmallGroup(128,906);
// by ID

G=gap.SmallGroup(128,906);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,436,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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