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## G = C4×C42.C2order 128 = 27

### Direct product of C4 and C42.C2

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C42.C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C42.C2
 Lower central C1 — C22 — C4×C42.C2
 Upper central C1 — C22×C4 — C4×C42.C2
 Jennings C1 — C23 — C4×C42.C2

Generators and relations for C4×C42.C2
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 332 in 242 conjugacy classes, 160 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×22], C22 [×3], C22 [×4], C2×C4 [×30], C2×C4 [×30], C23, C42 [×8], C42 [×18], C4⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×12], C2.C42 [×12], C2×C42 [×3], C2×C42 [×8], C2×C4⋊C4 [×12], C42.C2 [×8], C43, C4×C4⋊C4 [×6], C428C4, C23.63C23 [×4], C23.65C23 [×2], C2×C42.C2, C4×C42.C2
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×10], C24, C4×Q8 [×4], C42.C2 [×4], C23×C4, C22×Q8, C2×C4○D4 [×5], C2×C4×Q8, C4×C4○D4 [×2], C2×C42.C2, C23.36C23 [×2], C23.37C23, C4×C42.C2

Smallest permutation representation of C4×C42.C2
Regular action on 128 points
Generators in S128
(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 48 20 56)(2 45 17 53)(3 46 18 54)(4 47 19 55)(5 108 12 100)(6 105 9 97)(7 106 10 98)(8 107 11 99)(13 101 127 93)(14 102 128 94)(15 103 125 95)(16 104 126 96)(21 49 27 57)(22 50 28 58)(23 51 25 59)(24 52 26 60)(29 73 37 65)(30 74 38 66)(31 75 39 67)(32 76 40 68)(33 69 41 61)(34 70 42 62)(35 71 43 63)(36 72 44 64)(77 119 85 111)(78 120 86 112)(79 117 87 109)(80 118 88 110)(81 123 89 115)(82 124 90 116)(83 121 91 113)(84 122 92 114)
(1 42 22 40)(2 43 23 37)(3 44 24 38)(4 41 21 39)(5 110 14 116)(6 111 15 113)(7 112 16 114)(8 109 13 115)(9 119 125 121)(10 120 126 122)(11 117 127 123)(12 118 128 124)(17 35 25 29)(18 36 26 30)(19 33 27 31)(20 34 28 32)(45 63 51 65)(46 64 52 66)(47 61 49 67)(48 62 50 68)(53 71 59 73)(54 72 60 74)(55 69 57 75)(56 70 58 76)(77 103 83 105)(78 104 84 106)(79 101 81 107)(80 102 82 108)(85 95 91 97)(86 96 92 98)(87 93 89 99)(88 94 90 100)
(1 77 22 83)(2 78 23 84)(3 79 24 81)(4 80 21 82)(5 67 14 61)(6 68 15 62)(7 65 16 63)(8 66 13 64)(9 76 125 70)(10 73 126 71)(11 74 127 72)(12 75 128 69)(17 86 25 92)(18 87 26 89)(19 88 27 90)(20 85 28 91)(29 106 35 104)(30 107 36 101)(31 108 33 102)(32 105 34 103)(37 98 43 96)(38 99 44 93)(39 100 41 94)(40 97 42 95)(45 122 51 120)(46 123 52 117)(47 124 49 118)(48 121 50 119)(53 114 59 112)(54 115 60 109)(55 116 57 110)(56 113 58 111)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF 4AG ··· 4AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 Q8 C4○D4 kernel C4×C42.C2 C43 C4×C4⋊C4 C42⋊8C4 C23.63C23 C23.65C23 C2×C42.C2 C42.C2 C42 C2×C4 # reps 1 1 6 1 4 2 1 16 4 20

Matrix representation of C4×C42.C2 in GL5(𝔽5)

 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 4 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 1 0
,
 4 0 0 0 0 0 0 4 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 4

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[4,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,4] >;

C4×C42.C2 in GAP, Magma, Sage, TeX

C_4\times C_4^2.C_2
% in TeX

G:=Group("C4xC4^2.C2");
// GroupNames label

G:=SmallGroup(128,1037);
// by ID

G=gap.SmallGroup(128,1037);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,304]);
// Polycyclic

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

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