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## G = C2×C42⋊4C4order 128 = 27

### Direct product of C2 and C42⋊4C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C42⋊4C4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C2×C42⋊4C4
 Lower central C1 — C2 — C2×C42⋊4C4
 Upper central C1 — C23×C4 — C2×C42⋊4C4
 Jennings C1 — C23 — C2×C42⋊4C4

Generators and relations for C2×C424C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, cd=dc >

Subgroups: 540 in 420 conjugacy classes, 300 normal (6 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×24], C22 [×35], C2×C4 [×52], C2×C4 [×72], C23, C23 [×14], C42 [×48], C22×C4 [×50], C22×C4 [×24], C24, C2.C42 [×16], C2×C42 [×36], C23×C4, C23×C4 [×6], C2×C2.C42 [×4], C424C4 [×8], C22×C42 [×3], C2×C424C4
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C4○D4 [×8], C24, C2×C42 [×12], C42⋊C2 [×24], C23×C4 [×3], C2×C4○D4 [×4], C424C4 [×8], C22×C42, C2×C42⋊C2 [×6], C2×C424C4

Smallest permutation representation of C2×C424C4
Regular action on 128 points
Generators in S128
(1 40)(2 37)(3 38)(4 39)(5 127)(6 128)(7 125)(8 126)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(33 66)(34 67)(35 68)(36 65)(41 46)(42 47)(43 48)(44 45)(49 54)(50 55)(51 56)(52 53)(57 62)(58 63)(59 64)(60 61)(69 76)(70 73)(71 74)(72 75)(77 89)(78 90)(79 91)(80 92)(81 86)(82 87)(83 88)(84 85)(93 108)(94 105)(95 106)(96 107)(97 104)(98 101)(99 102)(100 103)(109 116)(110 113)(111 114)(112 115)(117 124)(118 121)(119 122)(120 123)
(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 7 36 102)(2 8 33 103)(3 5 34 104)(4 6 35 101)(9 63 95 78)(10 64 96 79)(11 61 93 80)(12 62 94 77)(13 59 107 91)(14 60 108 92)(15 57 105 89)(16 58 106 90)(17 87 111 55)(18 88 112 56)(19 85 109 53)(20 86 110 54)(21 83 115 51)(22 84 116 52)(23 81 113 49)(24 82 114 50)(25 75 119 46)(26 76 120 47)(27 73 117 48)(28 74 118 45)(29 69 123 42)(30 70 124 43)(31 71 121 44)(32 72 122 41)(37 126 66 100)(38 127 67 97)(39 128 68 98)(40 125 65 99)
(1 87 25 14)(2 56 26 105)(3 85 27 16)(4 54 28 107)(5 109 73 58)(6 20 74 91)(7 111 75 60)(8 18 76 89)(9 38 84 30)(10 68 81 121)(11 40 82 32)(12 66 83 123)(13 35 86 118)(15 33 88 120)(17 46 92 102)(19 48 90 104)(21 69 77 126)(22 43 78 97)(23 71 79 128)(24 41 80 99)(29 94 37 51)(31 96 39 49)(34 53 117 106)(36 55 119 108)(42 62 100 115)(44 64 98 113)(45 59 101 110)(47 57 103 112)(50 122 93 65)(52 124 95 67)(61 125 114 72)(63 127 116 70)

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

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

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

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 type + + + + image C1 C2 C2 C2 C4 C4○D4 kernel C2×C42⋊4C4 C2×C2.C42 C42⋊4C4 C22×C42 C2×C42 C23 # reps 1 4 8 3 48 16

Matrix representation of C2×C424C4 in GL5(𝔽5)

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

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

C2×C424C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_4C_4
% in TeX

G:=Group("C2xC4^2:4C4");
// GroupNames label

G:=SmallGroup(128,999);
// by ID

G=gap.SmallGroup(128,999);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,100]);
// Polycyclic

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

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