Copied to
clipboard

## G = C4×C4⋊Q8order 128 = 27

### Direct product of C4 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C4⋊Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C4⋊Q8
 Lower central C1 — C22 — C4×C4⋊Q8
 Upper central C1 — C22×C4 — C4×C4⋊Q8
 Jennings C1 — C23 — C4×C4⋊Q8

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

Subgroups: 412 in 294 conjugacy classes, 184 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C4 [×18], C22 [×3], C22 [×4], C2×C4 [×38], C2×C4 [×26], Q8 [×16], C23, C42 [×16], C42 [×14], C4⋊C4 [×16], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×8], C2×Q8 [×8], C2.C42 [×8], C2×C42 [×3], C2×C42 [×8], C2×C4⋊C4 [×14], C4×Q8 [×8], C4⋊Q8 [×8], C22×Q8 [×2], C43, C4×C4⋊C4 [×4], C429C4, C23.65C23 [×4], C23.67C23 [×2], C2×C4×Q8 [×2], C2×C4⋊Q8, C4×C4⋊Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×8], C23 [×15], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×6], C24, C4×D4 [×4], C4×Q8 [×8], C4⋊Q8 [×4], C23×C4, C22×D4, C22×Q8 [×2], C2×C4○D4 [×3], C2×C4×D4, C2×C4×Q8 [×2], C2×C4⋊Q8, C22.26C24, C23.37C23 [×2], C4×C4⋊Q8

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF 4AG ··· 4AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 Q8 C4○D4 kernel C4×C4⋊Q8 C43 C4×C4⋊C4 C42⋊9C4 C23.65C23 C23.67C23 C2×C4×Q8 C2×C4⋊Q8 C4⋊Q8 C42 C42 C2×C4 # reps 1 1 4 1 4 2 2 1 16 4 8 12

Matrix representation of C4×C4⋊Q8 in GL5(𝔽5)

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

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

C4×C4⋊Q8 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes Q_8
% in TeX

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

G:=SmallGroup(128,1039);
// by ID

G=gap.SmallGroup(128,1039);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184,304]);
// Polycyclic

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

׿
×
𝔽