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G = C4⋊Q32order 128 = 27

The semidirect product of C4 and Q32 acting via Q32/C16=C2

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

Aliases: C41Q32, C8.16D8, C16.8D4, C42.337D4, C4.8(C2×D8), (C4×C16).9C2, (C2×C4).84D8, C8.40(C2×D4), (C2×C8).253D4, (C2×Q32).3C2, C2.10(C2×Q32), C4.2(C41D4), C2.13(C84D4), (C2×C16).84C22, (C4×C8).402C22, (C2×C8).545C23, C4⋊Q16.10C2, C22.131(C2×D8), (C2×Q16).17C22, (C2×C4).813(C2×D4), SmallGroup(128,979)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4⋊Q32
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C4⋊Q32
C1C2C4C2×C8 — C4⋊Q32
C1C22C42C4×C8 — C4⋊Q32
C1C2C2C2C2C4C4C2×C8 — C4⋊Q32

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

Subgroups: 200 in 88 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C4 [×4], C22, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×8], C16 [×4], C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×12], C2×Q8 [×4], C4×C8, C2×C16 [×2], Q32 [×8], C4⋊Q8 [×2], C2×Q16 [×4], C2×Q16 [×2], C4×C16, C4⋊Q16 [×2], C2×Q32 [×4], C4⋊Q32
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], Q32 [×4], C41D4, C2×D8 [×2], C84D4, C2×Q32 [×2], C4⋊Q32

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

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

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

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

38 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J8A···8H16A···16P
order12224···444448···816···16
size11112···2161616162···22···2

38 irreducible representations

dim1111222222
type+++++++++-
imageC1C2C2C2D4D4D4D8D8Q32
kernelC4⋊Q32C4×C16C4⋊Q16C2×Q32C16C42C2×C8C8C2×C4C4
# reps11244114416

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

16200
16100
00160
00016
,
01100
31100
001311
00613
,
15900
11200
00512
001212
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,16,0,0,0,0,16],[0,3,0,0,11,11,0,0,0,0,13,6,0,0,11,13],[15,11,0,0,9,2,0,0,0,0,5,12,0,0,12,12] >;

C4⋊Q32 in GAP, Magma, Sage, TeX

C_4\rtimes Q_{32}
% in TeX

G:=Group("C4:Q32");
// GroupNames label

G:=SmallGroup(128,979);
// by ID

G=gap.SmallGroup(128,979);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,100,1123,360,4037,124]);
// Polycyclic

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

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