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## G = C4×C2.C42order 128 = 27

### Direct product of C4 and C2.C42

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×C2.C42
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C4×C2.C42
 Lower central C1 — C2 — C4×C2.C42
 Upper central C1 — C23×C4 — C4×C2.C42
 Jennings C1 — C24 — C4×C2.C42

Generators and relations for C4×C2.C42
G = < a,b,c,d | a4=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 532 in 376 conjugacy classes, 220 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C23×C4, C23×C4, C2×C2.C42, C22×C42, C4×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C43, C2×C2.C42, C424C4, C4×C22⋊C4, C4×C4⋊C4, C4×C2.C42

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

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

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

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

80 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 4Q ··· 4BL order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + - image C1 C2 C2 C4 C4 D4 Q8 C4○D4 kernel C4×C2.C42 C2×C2.C42 C22×C42 C2.C42 C2×C42 C22×C4 C22×C4 C23 # reps 1 4 3 32 24 6 2 8

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

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

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

C4×C2.C42 in GAP, Magma, Sage, TeX

C_4\times C_2.C_4^2
% in TeX

G:=Group("C4xC2.C4^2");
// GroupNames label

G:=SmallGroup(128,164);
// by ID

G=gap.SmallGroup(128,164);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,2,112,141,232,352]);
// Polycyclic

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

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