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G = (C2×Q8)⋊Q8order 128 = 27

1st semidirect product of C2×Q8 and Q8 acting via Q8/C2=C22

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

Aliases: (C2×Q8)⋊1Q8, C4⋊C4.83D4, (C2×C4).15Q16, (C2×C4).33SD16, C2.6(Q8⋊Q8), C2.6(C4.Q16), (C22×C4).312D4, C23.906(C2×D4), C4.30(C22⋊Q8), C22.54(C2×Q16), C4.31(C4.4D4), (C22×C8).66C22, C2.5(C4.SD16), C2.5(C23⋊Q8), C22.94(C2×SD16), C22.208C22≀C2, C2.20(C22⋊SD16), C22.4Q16.21C2, (C2×C42).354C22, C2.20(C22⋊Q16), (C22×Q8).60C22, C22.133(C8⋊C22), (C22×C4).1440C23, C22.87(C4.4D4), C22.103(C22⋊Q8), C22.122(C8.C22), C22.7C42.10C2, C23.67C23.13C2, C2.7(C42.28C22), (C2×C4⋊Q8).19C2, (C2×C4).280(C2×Q8), (C2×C4).1030(C2×D4), (C2×C4).617(C4○D4), (C2×C4⋊C4).113C22, (C2×Q8⋊C4).10C2, SmallGroup(128,756)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×Q8)⋊Q8
C1C2C22C2×C4C22×C4C22×Q8C23.67C23 — (C2×Q8)⋊Q8
C1C2C22×C4 — (C2×Q8)⋊Q8
C1C23C2×C42 — (C2×Q8)⋊Q8
C1C2C2C22×C4 — (C2×Q8)⋊Q8

Generators and relations for (C2×Q8)⋊Q8
 G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=abc, ede-1=d-1 >

Subgroups: 304 in 145 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×11], C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×14], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×13], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊Q8 [×4], C22×C8 [×2], C22×Q8, C22×Q8, C22.7C42, C22.4Q16 [×2], C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊Q8, (C2×Q8)⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×2], Q16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C2×SD16, C2×Q16, C8⋊C22, C8.C22, C23⋊Q8, C22⋊SD16, C22⋊Q16, Q8⋊Q8, C4.Q16, C4.SD16, C42.28C22, (C2×Q8)⋊Q8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111122222244
type++++++++--+-
imageC1C2C2C2C2C2D4D4Q8SD16Q16C4○D4C8⋊C22C8.C22
kernel(C2×Q8)⋊Q8C22.7C42C22.4Q16C23.67C23C2×Q8⋊C4C2×C4⋊Q8C4⋊C4C22×C4C2×Q8C2×C4C2×C4C2×C4C22C22
# reps11212142244611

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

1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
001000
000100
0000016
000010
,
1150000
0160000
001000
000100
00001212
0000125
,
750000
7100000
0010200
009700
000001
0000160
,
490000
0130000
004000
00111300
000040
0000013

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,12,5],[7,7,0,0,0,0,5,10,0,0,0,0,0,0,10,9,0,0,0,0,2,7,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,4,11,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13] >;

(C2×Q8)⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("(C2xQ8):Q8");
// GroupNames label

G:=SmallGroup(128,756);
// by ID

G=gap.SmallGroup(128,756);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,2804,718,172]);
// Polycyclic

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

׿
×
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