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## G = C4×C2.D8order 128 = 27

### Direct product of C4 and C2.D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C4×C2.D8
 Lower central C1 — C2 — C4 — C4×C2.D8
 Upper central C1 — C22×C4 — C2×C42 — C4×C2.D8
 Jennings C1 — C2 — C2 — C22×C4 — C4×C2.D8

Generators and relations for C4×C2.D8
G = < a,b,c,d | a4=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 140 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C2×C4×C8, C2×C2.D8, C4×C2.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C2.D8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×D8, C2×Q16, C4○D8, C4×C4⋊C4, C2×C2.D8, C23.25D4, C4×D8, C4×Q16, C4×C2.D8

Smallest permutation representation of C4×C2.D8
Regular action on 128 points
Generators in S128
(1 21 47 111)(2 22 48 112)(3 23 41 105)(4 24 42 106)(5 17 43 107)(6 18 44 108)(7 19 45 109)(8 20 46 110)(9 27 54 83)(10 28 55 84)(11 29 56 85)(12 30 49 86)(13 31 50 87)(14 32 51 88)(15 25 52 81)(16 26 53 82)(33 74 69 113)(34 75 70 114)(35 76 71 115)(36 77 72 116)(37 78 65 117)(38 79 66 118)(39 80 67 119)(40 73 68 120)(57 102 125 89)(58 103 126 90)(59 104 127 91)(60 97 128 92)(61 98 121 93)(62 99 122 94)(63 100 123 95)(64 101 124 96)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 113)(9 95)(10 96)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(49 103)(50 104)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)
(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 28 114 64)(2 27 115 63)(3 26 116 62)(4 25 117 61)(5 32 118 60)(6 31 119 59)(7 30 120 58)(8 29 113 57)(9 71 95 112)(10 70 96 111)(11 69 89 110)(12 68 90 109)(13 67 91 108)(14 66 92 107)(15 65 93 106)(16 72 94 105)(17 51 38 97)(18 50 39 104)(19 49 40 103)(20 56 33 102)(21 55 34 101)(22 54 35 100)(23 53 36 99)(24 52 37 98)(41 82 77 122)(42 81 78 121)(43 88 79 128)(44 87 80 127)(45 86 73 126)(46 85 74 125)(47 84 75 124)(48 83 76 123)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 Q8 D4 D8 Q16 C4○D4 C4○D8 kernel C4×C2.D8 C22.4Q16 C4×C4⋊C4 C2×C4×C8 C2×C2.D8 C4×C8 C2.D8 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 2 1 2 8 16 2 2 4 4 4 8

Matrix representation of C4×C2.D8 in GL5(𝔽17)

 13 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 13 0 0 0 0 0 13
,
 1 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 12 12 0 0 0 12 5 0 0 0 0 0 3 14 0 0 0 3 3
,
 1 0 0 0 0 0 5 12 0 0 0 12 12 0 0 0 0 0 0 13 0 0 0 13 0

G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,12,12,0,0,0,12,5,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,5,12,0,0,0,12,12,0,0,0,0,0,0,13,0,0,0,13,0] >;

C4×C2.D8 in GAP, Magma, Sage, TeX

C_4\times C_2.D_8
% in TeX

G:=Group("C4xC2.D8");
// GroupNames label

G:=SmallGroup(128,507);
// by ID

G=gap.SmallGroup(128,507);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,478,1018,248,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=b,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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