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## G = C4×C4.Q8order 128 = 27

### Direct product of C4 and C4.Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C4×C4.Q8
 Lower central C1 — C2 — C4 — C4×C4.Q8
 Upper central C1 — C22×C4 — C2×C42 — C4×C4.Q8
 Jennings C1 — C2 — C2 — C22×C4 — C4×C4.Q8

Generators and relations for C4×C4.Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 228 in 140 conjugacy classes, 92 normal (18 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, C4.Q8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22.4Q16, C4×C4⋊C4, C2×C4×C8, C2×C4.Q8, C4×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×SD16, C4○D8, C4×C4⋊C4, C2×C4.Q8, C23.25D4, C4×SD16, C4×C4.Q8

Smallest permutation representation of C4×C4.Q8
Regular action on 128 points
Generators in S128
(1 18 43 25)(2 19 44 26)(3 20 45 27)(4 21 46 28)(5 22 47 29)(6 23 48 30)(7 24 41 31)(8 17 42 32)(9 97 59 81)(10 98 60 82)(11 99 61 83)(12 100 62 84)(13 101 63 85)(14 102 64 86)(15 103 57 87)(16 104 58 88)(33 51 89 124)(34 52 90 125)(35 53 91 126)(36 54 92 127)(37 55 93 128)(38 56 94 121)(39 49 95 122)(40 50 96 123)(65 112 80 120)(66 105 73 113)(67 106 74 114)(68 107 75 115)(69 108 76 116)(70 109 77 117)(71 110 78 118)(72 111 79 119)
(1 75 5 79)(2 76 6 80)(3 77 7 73)(4 78 8 74)(9 51 13 55)(10 52 14 56)(11 53 15 49)(12 54 16 50)(17 114 21 118)(18 115 22 119)(19 116 23 120)(20 117 24 113)(25 107 29 111)(26 108 30 112)(27 109 31 105)(28 110 32 106)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)(41 66 45 70)(42 67 46 71)(43 68 47 72)(44 69 48 65)(57 122 61 126)(58 123 62 127)(59 124 63 128)(60 125 64 121)(89 101 93 97)(90 102 94 98)(91 103 95 99)(92 104 96 100)
(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 64 73 127)(2 59 74 122)(3 62 75 125)(4 57 76 128)(5 60 77 123)(6 63 78 126)(7 58 79 121)(8 61 80 124)(9 67 49 44)(10 70 50 47)(11 65 51 42)(12 68 52 45)(13 71 53 48)(14 66 54 43)(15 69 55 46)(16 72 56 41)(17 83 120 33)(18 86 113 36)(19 81 114 39)(20 84 115 34)(21 87 116 37)(22 82 117 40)(23 85 118 35)(24 88 119 38)(25 102 105 92)(26 97 106 95)(27 100 107 90)(28 103 108 93)(29 98 109 96)(30 101 110 91)(31 104 111 94)(32 99 112 89)

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

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

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

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 type + + + + + - + image C1 C2 C2 C2 C2 C4 C4 Q8 D4 SD16 C4○D4 C4○D8 kernel C4×C4.Q8 C22.4Q16 C4×C4⋊C4 C2×C4×C8 C2×C4.Q8 C4×C8 C4.Q8 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 2 2 1 2 8 16 2 2 8 4 8

Matrix representation of C4×C4.Q8 in GL4(𝔽17) generated by

 4 0 0 0 0 16 0 0 0 0 13 0 0 0 0 13
,
 16 0 0 0 0 16 0 0 0 0 0 16 0 0 1 0
,
 16 0 0 0 0 16 0 0 0 0 12 12 0 0 5 12
,
 13 0 0 0 0 4 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [4,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,16,0,0,0,0,12,5,0,0,12,12],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1] >;

C4×C4.Q8 in GAP, Magma, Sage, TeX

C_4\times C_4.Q_8
% in TeX

G:=Group("C4xC4.Q8");
// GroupNames label

G:=SmallGroup(128,506);
// by ID

G=gap.SmallGroup(128,506);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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