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G = (C2×C8)⋊Q8order 128 = 27

6th semidirect product of C2×C8 and Q8 acting via Q8/C2=C22

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

Aliases: C4⋊C42Q8, (C2×C8)⋊6Q8, C2.7(C8⋊Q8), C4⋊C4.100D4, C4.22(C4⋊Q8), C2.4(C83Q8), (C2×C4).36SD16, C4.7(C22⋊Q8), C2.7(Q8⋊Q8), C2.7(D42Q8), (C22×C4).316D4, C23.920(C2×D4), C22.49(C4⋊Q8), C2.23(Q8⋊D4), C2.23(C22⋊SD16), C22.228C22≀C2, C22.4Q16.43C2, (C2×C42).372C22, (C22×C8).323C22, C22.100(C2×SD16), C22.143(C8⋊C22), (C22×C4).1454C23, C22.107(C22⋊Q8), C22.132(C8.C22), C22.7C42.35C2, C23.65C23.16C2, C2.5(C23.78C23), (C2×C4⋊Q8).22C2, (C2×C4).218(C2×Q8), (C2×C4.Q8).22C2, (C2×C4).1049(C2×D4), (C2×C4).776(C4○D4), (C2×C4⋊C4).135C22, SmallGroup(128,790)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C8)⋊Q8
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.65C23 — (C2×C8)⋊Q8
C1C2C22×C4 — (C2×C8)⋊Q8
C1C23C2×C42 — (C2×C8)⋊Q8
C1C2C2C22×C4 — (C2×C8)⋊Q8

Generators and relations for (C2×C8)⋊Q8
 G = < a,b,c,d | a2=b8=c4=1, d2=c2, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b3, dcd-1=c-1 >

Subgroups: 288 in 141 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×11], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×8], C23, C42 [×2], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42, C4.Q8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16 [×2], C23.65C23, C2×C4.Q8 [×2], C2×C4⋊Q8, (C2×C8)⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, SD16 [×4], C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C2×SD16 [×2], C8⋊C22, C8.C22, C23.78C23, Q8⋊D4, C22⋊SD16, Q8⋊Q8, D42Q8, C83Q8, C8⋊Q8, (C2×C8)⋊Q8

Smallest permutation representation of (C2×C8)⋊Q8
Regular action on 128 points
Generators in S128
(1 117)(2 118)(3 119)(4 120)(5 113)(6 114)(7 115)(8 116)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(65 110)(66 111)(67 112)(68 105)(69 106)(70 107)(71 108)(72 109)(73 100)(74 101)(75 102)(76 103)(77 104)(78 97)(79 98)(80 99)(81 92)(82 93)(83 94)(84 95)(85 96)(86 89)(87 90)(88 91)
(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 16 53 106)(2 11 54 109)(3 14 55 112)(4 9 56 107)(5 12 49 110)(6 15 50 105)(7 10 51 108)(8 13 52 111)(17 89 100 30)(18 92 101 25)(19 95 102 28)(20 90 103 31)(21 93 104 26)(22 96 97 29)(23 91 98 32)(24 94 99 27)(33 41 84 75)(34 44 85 78)(35 47 86 73)(36 42 87 76)(37 45 88 79)(38 48 81 74)(39 43 82 77)(40 46 83 80)(57 126 66 116)(58 121 67 119)(59 124 68 114)(60 127 69 117)(61 122 70 120)(62 125 71 115)(63 128 72 118)(64 123 65 113)
(1 103 53 20)(2 77 54 43)(3 97 55 22)(4 79 56 45)(5 99 49 24)(6 73 50 47)(7 101 51 18)(8 75 52 41)(9 88 107 37)(10 92 108 25)(11 82 109 39)(12 94 110 27)(13 84 111 33)(14 96 112 29)(15 86 105 35)(16 90 106 31)(17 114 100 124)(19 116 102 126)(21 118 104 128)(23 120 98 122)(26 63 93 72)(28 57 95 66)(30 59 89 68)(32 61 91 70)(34 58 85 67)(36 60 87 69)(38 62 81 71)(40 64 83 65)(42 117 76 127)(44 119 78 121)(46 113 80 123)(48 115 74 125)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim11111122222244
type+++++++--++-
imageC1C2C2C2C2C2D4Q8Q8D4SD16C4○D4C8⋊C22C8.C22
kernel(C2×C8)⋊Q8C22.7C42C22.4Q16C23.65C23C2×C4.Q8C2×C4⋊Q8C4⋊C4C4⋊C4C2×C8C22×C4C2×C4C2×C4C22C22
# reps11212142428211

Matrix representation of (C2×C8)⋊Q8 in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
1070000
500000
0001300
0013000
0000104
0000137
,
400000
4130000
004000
0001300
0000410
0000713
,
1620000
1610000
000400
004000
0000137
0000104

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,7,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,10,13,0,0,0,0,4,7],[4,4,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,7,0,0,0,0,10,13],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,13,10,0,0,0,0,7,4] >;

(C2×C8)⋊Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes Q_8
% in TeX

G:=Group("(C2xC8):Q8");
// GroupNames label

G:=SmallGroup(128,790);
// by ID

G=gap.SmallGroup(128,790);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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