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## G = Q8×C4⋊C4order 128 = 27

### Direct product of Q8 and C4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C4⋊C4
G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 396 in 284 conjugacy classes, 196 normal (21 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C4×C4⋊C4, C429C4, C23.65C23, C2×C4×Q8, C2×C4×Q8, Q8×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C4⋊C4, C2×C4×Q8, C23.33C23, D4×Q8, Q86D4, Q83Q8, Q82, Q8×C4⋊C4

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

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4AP order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + - + - + - image C1 C2 C2 C2 C2 C4 Q8 D4 Q8 C4○D4 2+ 1+4 2- 1+4 kernel Q8×C4⋊C4 C4×C4⋊C4 C42⋊9C4 C23.65C23 C2×C4×Q8 C4×Q8 C4⋊C4 C2×Q8 C2×Q8 C2×C4 C22 C22 # reps 1 3 3 6 3 16 4 4 4 4 1 1

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

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

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

Q8×C4⋊C4 in GAP, Magma, Sage, TeX

Q_8\times C_4\rtimes C_4
% in TeX

G:=Group("Q8xC4:C4");
// GroupNames label

G:=SmallGroup(128,1082);
// by ID

G=gap.SmallGroup(128,1082);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184,346,80]);
// Polycyclic

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