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G = Q8⋊(C4⋊C4)  order 128 = 27

4th semidirect product of Q8 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

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

Aliases: Q84(C4⋊C4), C4.54(C4×D4), C2.6(C4×Q16), C4⋊C4.305D4, Q8⋊C43C4, (C2×Q8).20Q8, (C2×C4).47Q16, (C2×Q8).214D4, C2.2(Q8.Q8), C2.3(D4⋊D4), C2.2(C4.Q16), C23.761(C2×D4), (C22×C4).684D4, C22.145(C4×D4), C4.26(C22⋊Q8), C22.30(C2×Q16), C22.85C22≀C2, C22.50(C4○D8), C22.4Q16.5C2, C2.3(C22⋊Q16), C2.9(SD16⋊C4), C22.71(C8⋊C22), (C2×C42).269C22, (C22×C8).103C22, C4.7(C22.D4), C22.72(C22⋊Q8), (C22×C4).1354C23, C22.60(C8.C22), C2.19(C23.8Q8), (C22×Q8).385C22, C23.65C23.2C2, C22.7C42.18C2, C4.13(C2×C4⋊C4), (C2×C4×Q8).13C2, C4⋊C4.65(C2×C4), (C2×C8).35(C2×C4), (C2×C2.D8).4C2, (C2×C4).989(C2×D4), (C2×C4).265(C2×Q8), (C2×Q8).143(C2×C4), (C2×C4).750(C4○D4), (C2×C4⋊C4).758C22, (C2×C4).372(C22×C4), (C2×Q8⋊C4).19C2, SmallGroup(128,595)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8⋊(C4⋊C4)
C1C2C22C2×C4C22×C4C22×Q8C2×C4×Q8 — Q8⋊(C4⋊C4)
C1C2C2×C4 — Q8⋊(C4⋊C4)
C1C23C2×C42 — Q8⋊(C4⋊C4)
C1C2C2C22×C4 — Q8⋊(C4⋊C4)

Generators and relations for Q8⋊(C4⋊C4)
 G = < a,b,c,d | a4=c4=d4=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 268 in 149 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×12], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×22], Q8 [×4], Q8 [×6], C23, C42 [×6], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×3], C2.C42, Q8⋊C4 [×4], Q8⋊C4 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4×Q8 [×4], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16, C23.65C23, C2×Q8⋊C4 [×2], C2×C2.D8, C2×C4×Q8, Q8⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], Q16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.8Q8, C4×Q16, SD16⋊C4, D4⋊D4, C22⋊Q16, C4.Q16, Q8.Q8, 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 24 3 22)(2 23 4 21)(5 115 7 113)(6 114 8 116)(9 28 11 26)(10 27 12 25)(13 29 15 31)(14 32 16 30)(17 35 19 33)(18 34 20 36)(37 53 39 55)(38 56 40 54)(41 60 43 58)(42 59 44 57)(45 61 47 63)(46 64 48 62)(49 68 51 66)(50 67 52 65)(69 88 71 86)(70 87 72 85)(73 91 75 89)(74 90 76 92)(77 96 79 94)(78 95 80 93)(81 100 83 98)(82 99 84 97)(101 120 103 118)(102 119 104 117)(105 123 107 121)(106 122 108 124)(109 128 111 126)(110 127 112 125)
(1 5 13 121)(2 8 14 124)(3 7 15 123)(4 6 16 122)(9 126 19 117)(10 125 20 120)(11 128 17 119)(12 127 18 118)(21 113 30 107)(22 116 31 106)(23 115 32 105)(24 114 29 108)(25 111 34 104)(26 110 35 103)(27 109 36 102)(28 112 33 101)(37 95 46 88)(38 94 47 87)(39 93 48 86)(40 96 45 85)(41 98 51 91)(42 97 52 90)(43 100 49 89)(44 99 50 92)(53 79 64 70)(54 78 61 69)(55 77 62 72)(56 80 63 71)(57 83 67 73)(58 82 68 76)(59 81 65 75)(60 84 66 74)
(1 58 11 53)(2 59 12 54)(3 60 9 55)(4 57 10 56)(5 76 128 70)(6 73 125 71)(7 74 126 72)(8 75 127 69)(13 68 17 64)(14 65 18 61)(15 66 19 62)(16 67 20 63)(21 44 27 38)(22 41 28 39)(23 42 25 40)(24 43 26 37)(29 49 35 46)(30 50 36 47)(31 51 33 48)(32 52 34 45)(77 123 84 117)(78 124 81 118)(79 121 82 119)(80 122 83 120)(85 115 90 111)(86 116 91 112)(87 113 92 109)(88 114 89 110)(93 106 98 101)(94 107 99 102)(95 108 100 103)(96 105 97 104)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim11111111222222244
type++++++++++--+-
imageC1C2C2C2C2C2C2C4D4D4D4Q8Q16C4○D4C4○D8C8⋊C22C8.C22
kernelQ8⋊(C4⋊C4)C22.7C42C22.4Q16C23.65C23C2×Q8⋊C4C2×C2.D8C2×C4×Q8Q8⋊C4C4⋊C4C22×C4C2×Q8C2×Q8C2×C4C2×C4C22C22C22
# reps11112118222244411

Matrix representation of Q8⋊(C4⋊C4) in GL5(𝔽17)

10000
01000
00100
0001615
00011
,
160000
016000
001600
000130
00044
,
160000
013800
00400
00007
000120
,
130000
016200
016100
000139
00044

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,4,0,0,0,0,4],[16,0,0,0,0,0,13,0,0,0,0,8,4,0,0,0,0,0,0,12,0,0,0,7,0],[13,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,13,4,0,0,0,9,4] >;

Q8⋊(C4⋊C4) in GAP, Magma, Sage, TeX

Q_8\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("Q8:(C4:C4)");
// GroupNames label

G:=SmallGroup(128,595);
// by ID

G=gap.SmallGroup(128,595);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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