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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

Aliases: Q8×C4⋊C4, C23.232C24, C22.492- 1+4, C22.682+ 1+4, C2.1Q82, C43(C4×Q8), C2.3(D4×Q8), (C4×Q8)⋊19C4, (C2×Q8).26Q8, (C2×Q8).260D4, C42.186(C2×C4), C2.1(Q83Q8), C2.2(Q86D4), C429C4.23C2, C22.40(C22×Q8), C22.123(C23×C4), (C2×C42).431C22, C22.107(C22×D4), (C22×C4).1247C23, (C22×Q8).509C22, C2.C42.521C22, C23.65C23.33C2, C2.28(C23.33C23), C4.18(C2×C4⋊C4), C2.16(C2×C4×Q8), (C4×C4⋊C4).38C2, (C2×C4×Q8).23C2, C4⋊C4.208(C2×C4), C2.16(C22×C4⋊C4), (C2×C4).301(C2×Q8), (C2×C4).1071(C2×D4), (C2×Q8).219(C2×C4), (C2×C4).889(C4○D4), (C2×C4⋊C4).821C22, (C2×C4).567(C22×C4), C22.117(C2×C4○D4), SmallGroup(128,1082)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q8×C4⋊C4
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — Q8×C4⋊C4
C1C22 — Q8×C4⋊C4
C1C23 — Q8×C4⋊C4
C1C23 — 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 [×7], C4 [×16], C4 [×16], C22 [×7], C2×C4 [×34], C2×C4 [×28], Q8 [×16], C23, C42 [×12], C42 [×12], C4⋊C4 [×16], C4⋊C4 [×24], C22×C4, C22×C4 [×14], C2×Q8 [×12], C2.C42 [×6], C2×C42 [×9], C2×C4⋊C4, C2×C4⋊C4 [×18], C4×Q8 [×8], C4×Q8 [×8], C22×Q8, C4×C4⋊C4 [×3], C429C4 [×3], C23.65C23 [×6], C2×C4×Q8, C2×C4×Q8 [×2], Q8×C4⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×8], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×2], C24, C2×C4⋊C4 [×12], C4×Q8 [×4], C23×C4, C22×D4, C22×Q8 [×2], 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 53 3 55)(2 56 4 54)(5 93 7 95)(6 96 8 94)(9 99 11 97)(10 98 12 100)(13 101 15 103)(14 104 16 102)(17 48 19 46)(18 47 20 45)(21 60 23 58)(22 59 24 57)(25 50 27 52)(26 49 28 51)(29 76 31 74)(30 75 32 73)(33 68 35 66)(34 67 36 65)(37 70 39 72)(38 69 40 71)(41 62 43 64)(42 61 44 63)(77 122 79 124)(78 121 80 123)(81 116 83 114)(82 115 84 113)(85 118 87 120)(86 117 88 119)(89 112 91 110)(90 111 92 109)(105 128 107 126)(106 127 108 125)
(1 37 19 30)(2 38 20 31)(3 39 17 32)(4 40 18 29)(5 111 14 118)(6 112 15 119)(7 109 16 120)(8 110 13 117)(9 115 127 124)(10 116 128 121)(11 113 125 122)(12 114 126 123)(21 44 28 35)(22 41 25 36)(23 42 26 33)(24 43 27 34)(45 74 56 69)(46 75 53 70)(47 76 54 71)(48 73 55 72)(49 68 58 61)(50 65 59 62)(51 66 60 63)(52 67 57 64)(77 99 84 108)(78 100 81 105)(79 97 82 106)(80 98 83 107)(85 95 90 102)(86 96 91 103)(87 93 92 104)(88 94 89 101)
(1 90 22 82)(2 91 23 83)(3 92 24 84)(4 89 21 81)(5 62 9 70)(6 63 10 71)(7 64 11 72)(8 61 12 69)(13 68 126 74)(14 65 127 75)(15 66 128 76)(16 67 125 73)(17 87 27 77)(18 88 28 78)(19 85 25 79)(20 86 26 80)(29 101 35 105)(30 102 36 106)(31 103 33 107)(32 104 34 108)(37 95 41 97)(38 96 42 98)(39 93 43 99)(40 94 44 100)(45 117 49 123)(46 118 50 124)(47 119 51 121)(48 120 52 122)(53 111 59 115)(54 112 60 116)(55 109 57 113)(56 110 58 114)

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

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

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

50 conjugacy classes

class 1 2A···2G4A···4X4Y···4AP
order12···24···44···4
size11···12···24···4

50 irreducible representations

dim111111222244
type+++++-+-+-
imageC1C2C2C2C2C4Q8D4Q8C4○D42+ 1+42- 1+4
kernelQ8×C4⋊C4C4×C4⋊C4C429C4C23.65C23C2×C4×Q8C4×Q8C4⋊C4C2×Q8C2×Q8C2×C4C22C22
# reps1336316444411

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

10000
01000
00100
00020
00033
,
40000
01000
00100
00012
00044
,
10000
02000
01300
00010
00001
,
20000
03300
00200
00010
00001

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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