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## G = C22×Q32order 128 = 27

### Direct product of C22 and Q32

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C22×Q32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×Q16 — C22×Q32
 Lower central C1 — C2 — C4 — C8 — C22×Q32
 Upper central C1 — C23 — C22×C4 — C22×C8 — C22×Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C22×Q32

Generators and relations for C22×Q32
G = < a,b,c,d | a2=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 340 in 180 conjugacy classes, 100 normal (9 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C8, C8 [×3], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, C16 [×4], C2×C8 [×6], Q16 [×8], Q16 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×18], C2×C16 [×6], Q32 [×16], C22×C8, C2×Q16 [×12], C2×Q16 [×6], C22×Q8 [×2], C22×C16, C2×Q32 [×12], C22×Q16 [×2], C22×Q32
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, Q32 [×4], C2×D8 [×6], C22×D4, C2×Q32 [×6], C22×D8, C22×Q32

Smallest permutation representation of C22×Q32
Regular action on 128 points
Generators in S128
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 97)(15 98)(16 99)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 65)(30 66)(31 67)(32 68)(33 64)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(81 126)(82 127)(83 128)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 113)(32 114)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 97)(48 98)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 81)
(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 94 9 86)(2 93 10 85)(3 92 11 84)(4 91 12 83)(5 90 13 82)(6 89 14 81)(7 88 15 96)(8 87 16 95)(17 34 25 42)(18 33 26 41)(19 48 27 40)(20 47 28 39)(21 46 29 38)(22 45 30 37)(23 44 31 36)(24 43 32 35)(49 77 57 69)(50 76 58 68)(51 75 59 67)(52 74 60 66)(53 73 61 65)(54 72 62 80)(55 71 63 79)(56 70 64 78)(97 126 105 118)(98 125 106 117)(99 124 107 116)(100 123 108 115)(101 122 109 114)(102 121 110 113)(103 120 111 128)(104 119 112 127)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 D4 D4 D8 D8 Q32 kernel C22×Q32 C22×C16 C2×Q32 C22×Q16 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 1 12 2 3 1 6 2 16

Matrix representation of C22×Q32 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 11 0 0 0 6 1 0 0 0 0 0 7 12 0 0 0 11 2
,
 1 0 0 0 0 0 0 16 0 0 0 16 0 0 0 0 0 0 7 10 0 0 0 12 10

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,6,0,0,0,11,1,0,0,0,0,0,7,11,0,0,0,12,2],[1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,7,12,0,0,0,10,10] >;

C22×Q32 in GAP, Magma, Sage, TeX

C_2^2\times Q_{32}
% in TeX

G:=Group("C2^2xQ32");
// GroupNames label

G:=SmallGroup(128,2142);
// by ID

G=gap.SmallGroup(128,2142);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,456,1684,851,242,4037,2028,124]);
// 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=c^-1>;
// generators/relations

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