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## G = Q8×C42order 128 = 27

### Direct product of C42 and Q8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C42
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 396 in 342 conjugacy classes, 288 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×24], C4 [×18], C22, C22 [×6], C2×C4 [×54], C2×C4 [×18], Q8 [×16], C23, C42 [×40], C42 [×18], C4⋊C4 [×36], C22×C4 [×15], C2×Q8 [×12], C2.C42 [×6], C2×C42, C2×C42 [×18], C2×C4⋊C4 [×9], C4×Q8 [×24], C22×Q8, C43 [×3], C4×C4⋊C4 [×9], C2×C4×Q8 [×3], Q8×C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], Q8 [×4], C23 [×15], C42 [×16], C22×C4 [×42], C2×Q8 [×6], C4○D4 [×6], C24, C2×C42 [×12], C4×Q8 [×12], C23×C4 [×3], C22×Q8, C2×C4○D4 [×3], C22×C42, C2×C4×Q8 [×3], C4×C4○D4 [×3], Q8×C42

Smallest permutation representation of Q8×C42
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 39 31 15)(2 40 32 16)(3 37 29 13)(4 38 30 14)(5 20 121 106)(6 17 122 107)(7 18 123 108)(8 19 124 105)(9 49 33 28)(10 50 34 25)(11 51 35 26)(12 52 36 27)(21 127 111 117)(22 128 112 118)(23 125 109 119)(24 126 110 120)(41 81 65 60)(42 82 66 57)(43 83 67 58)(44 84 68 59)(45 55 69 61)(46 56 70 62)(47 53 71 63)(48 54 72 64)(73 113 97 92)(74 114 98 89)(75 115 99 90)(76 116 100 91)(77 87 101 93)(78 88 102 94)(79 85 103 95)(80 86 104 96)
(1 43 10 54)(2 44 11 55)(3 41 12 56)(4 42 9 53)(5 102 21 113)(6 103 22 114)(7 104 23 115)(8 101 24 116)(13 60 27 46)(14 57 28 47)(15 58 25 48)(16 59 26 45)(17 95 128 98)(18 96 125 99)(19 93 126 100)(20 94 127 97)(29 65 36 62)(30 66 33 63)(31 67 34 64)(32 68 35 61)(37 81 52 70)(38 82 49 71)(39 83 50 72)(40 84 51 69)(73 106 88 117)(74 107 85 118)(75 108 86 119)(76 105 87 120)(77 110 91 124)(78 111 92 121)(79 112 89 122)(80 109 90 123)
(1 114 10 103)(2 115 11 104)(3 116 12 101)(4 113 9 102)(5 42 21 53)(6 43 22 54)(7 44 23 55)(8 41 24 56)(13 76 27 87)(14 73 28 88)(15 74 25 85)(16 75 26 86)(17 83 128 72)(18 84 125 69)(19 81 126 70)(20 82 127 71)(29 91 36 77)(30 92 33 78)(31 89 34 79)(32 90 35 80)(37 100 52 93)(38 97 49 94)(39 98 50 95)(40 99 51 96)(45 108 59 119)(46 105 60 120)(47 106 57 117)(48 107 58 118)(61 123 68 109)(62 124 65 110)(63 121 66 111)(64 122 67 112)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4BT 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 2 2 type + + + + - image C1 C2 C2 C2 C4 Q8 C4○D4 kernel Q8×C42 C43 C4×C4⋊C4 C2×C4×Q8 C4×Q8 C42 C2×C4 # reps 1 3 9 3 48 4 12

Matrix representation of Q8×C42 in GL4(𝔽5) generated by

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

Q8×C42 in GAP, Magma, Sage, TeX

Q_8\times C_4^2
% in TeX

G:=Group("Q8xC4^2");
// GroupNames label

G:=SmallGroup(128,1004);
// by ID

G=gap.SmallGroup(128,1004);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,436,192]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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