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## G = C2×C4×C4⋊C4order 128 = 27

### Direct product of C2×C4 and C4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C4×C4⋊C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 588 in 464 conjugacy classes, 340 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C4 [×16], C4 [×20], C22 [×3], C22 [×32], C2×C4 [×76], C2×C4 [×60], C23, C23 [×14], C42 [×16], C42 [×16], C4⋊C4 [×32], C22×C4 [×58], C22×C4 [×20], C24, C2.C42 [×8], C2×C42 [×20], C2×C42 [×8], C2×C4⋊C4 [×24], C23×C4 [×3], C23×C4 [×4], C2×C2.C42 [×2], C4×C4⋊C4 [×8], C22×C42, C22×C42 [×2], C22×C4⋊C4 [×2], C2×C4×C4⋊C4
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], D4 [×4], Q8 [×4], C23 [×15], C42 [×16], C4⋊C4 [×16], C22×C4 [×42], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×4], C24, C2×C42 [×12], C2×C4⋊C4 [×12], C42⋊C2 [×4], C4×D4 [×8], C4×Q8 [×8], C23×C4 [×3], C22×D4, C22×Q8, C2×C4○D4 [×2], C4×C4⋊C4 [×8], C22×C42, C22×C4⋊C4, C2×C42⋊C2, C2×C4×D4 [×2], C2×C4×Q8 [×2], C2×C4×C4⋊C4

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

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

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

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

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 1 1 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 C4○D4 kernel C2×C4×C4⋊C4 C2×C2.C42 C4×C4⋊C4 C22×C42 C22×C4⋊C4 C2×C42 C2×C4⋊C4 C22×C4 C22×C4 C23 # reps 1 2 8 3 2 16 32 4 4 8

Matrix representation of C2×C4×C4⋊C4 in GL5(𝔽5)

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

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

C2×C4×C4⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_4\rtimes C_4
% in TeX

G:=Group("C2xC4xC4:C4");
// GroupNames label

G:=SmallGroup(128,1001);
// by ID

G=gap.SmallGroup(128,1001);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,268]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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