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G = Q8⋊C4⋊C4order 128 = 27

6th semidirect product of Q8⋊C4 and C4 acting via C4/C2=C2

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

Aliases: Q85(C4⋊C4), C4.56(C4×D4), C4⋊C4.307D4, Q8⋊C46C4, (C2×Q8).21Q8, C2.9(C4×SD16), (C2×Q8).215D4, C2.3(Q8.Q8), C2.3(Q8⋊D4), C2.2(Q8⋊Q8), (C2×C4).110SD16, C22.147(C4×D4), (C22×C4).686D4, C23.763(C2×D4), C4.28(C22⋊Q8), C2.6(Q16⋊C4), C22.87C22≀C2, C2.4(D4.7D4), C22.52(C4○D8), C22.4Q16.6C2, C22.55(C2×SD16), (C22×C8).314C22, (C2×C42).271C22, C4.9(C22.D4), C22.74(C22⋊Q8), (C22×C4).1356C23, C22.61(C8.C22), C2.21(C23.8Q8), (C22×Q8).386C22, C22.7C42.31C2, C23.65C23.3C2, C4.15(C2×C4⋊C4), (C2×C4×Q8).14C2, C4⋊C4.66(C2×C4), (C2×C8).107(C2×C4), (C2×C4).991(C2×D4), (C2×C4).267(C2×Q8), (C2×C4.Q8).15C2, (C2×Q8).144(C2×C4), (C2×C4).752(C4○D4), (C2×C4⋊C4).760C22, (C2×C4).374(C22×C4), (C2×Q8⋊C4).27C2, SmallGroup(128,597)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111122222224
type++++++++++--
imageC1C2C2C2C2C2C2C4D4D4D4Q8SD16C4○D4C4○D8C8.C22
kernelQ8⋊C4⋊C4C22.7C42C22.4Q16C23.65C23C2×Q8⋊C4C2×C4.Q8C2×C4×Q8Q8⋊C4C4⋊C4C22×C4C2×Q8C2×Q8C2×C4C2×C4C22C22
# reps1111211822224442

Matrix representation of Q8⋊C4⋊C4 in GL6(𝔽17)

1150000
1160000
000100
0016000
000010
000001
,
1070000
570000
000400
004000
000010
000001
,
100000
1160000
003300
0031400
0000159
000072
,
7100000
12100000
0001600
0016000
0000815
000079

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,15,7,0,0,0,0,9,2],[7,12,0,0,0,0,10,10,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,8,7,0,0,0,0,15,9] >;

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(128,597);
// by ID

G=gap.SmallGroup(128,597);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,352,2019,1018,521,248,2804,1411,718,172]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

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