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## G = C4×Q32order 128 = 27

### Direct product of C4 and Q32

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C4×Q32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×Q16 — C4×Q32
 Lower central C1 — C2 — C4 — C8 — C4×Q32
 Upper central C1 — C2×C4 — C42 — C4×C8 — C4×Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C16, C16, C42, C42, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, C4×C8, Q8⋊C4, C2.D8, C2×C16, Q32, C4×Q8, C2×Q16, C4×C16, C2.Q32, C163C4, C4×Q16, C2×Q32, C4×Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, Q32, 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
(1 107 35 87)(2 108 36 88)(3 109 37 89)(4 110 38 90)(5 111 39 91)(6 112 40 92)(7 97 41 93)(8 98 42 94)(9 99 43 95)(10 100 44 96)(11 101 45 81)(12 102 46 82)(13 103 47 83)(14 104 48 84)(15 105 33 85)(16 106 34 86)(17 60 123 69)(18 61 124 70)(19 62 125 71)(20 63 126 72)(21 64 127 73)(22 49 128 74)(23 50 113 75)(24 51 114 76)(25 52 115 77)(26 53 116 78)(27 54 117 79)(28 55 118 80)(29 56 119 65)(30 57 120 66)(31 58 121 67)(32 59 122 68)
(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 67 9 75)(2 66 10 74)(3 65 11 73)(4 80 12 72)(5 79 13 71)(6 78 14 70)(7 77 15 69)(8 76 16 68)(17 97 25 105)(18 112 26 104)(19 111 27 103)(20 110 28 102)(21 109 29 101)(22 108 30 100)(23 107 31 99)(24 106 32 98)(33 60 41 52)(34 59 42 51)(35 58 43 50)(36 57 44 49)(37 56 45 64)(38 55 46 63)(39 54 47 62)(40 53 48 61)(81 127 89 119)(82 126 90 118)(83 125 91 117)(84 124 92 116)(85 123 93 115)(86 122 94 114)(87 121 95 113)(88 120 96 128)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H 16A ··· 16P order 1 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 16 ··· 16 size 1 1 1 1 1 1 1 1 2 2 2 2 8 ··· 8 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 D8 Q32 C4○D8 C4○D16 kernel C4×Q32 C4×C16 C2.Q32 C16⋊3C4 C4×Q16 C2×Q32 Q32 C42 C2×C8 C8 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 1 2 4 8 4 8

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

 4 0 0 0 0 4 0 0 0 0 16 0 0 0 0 16
,
 14 3 0 0 14 14 0 0 0 0 6 13 0 0 4 6
,
 14 3 0 0 3 3 0 0 0 0 12 5 0 0 5 5
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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