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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×Q8)⋊Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C23.67C23 — (C2×Q8)⋊Q8
 Lower central C1 — C2 — C22×C4 — (C2×Q8)⋊Q8
 Upper central C1 — C23 — C2×C42 — (C2×Q8)⋊Q8
 Jennings C1 — C2 — C2 — C22×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, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C22×Q8, C22×Q8, C22.7C42, C22.4Q16, C23.67C23, C2×Q8⋊C4, C2×C4⋊Q8, (C2×Q8)⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, SD16, Q16, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4.4D4, 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 ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 D4 D4 Q8 SD16 Q16 C4○D4 C8⋊C22 C8.C22 kernel (C2×Q8)⋊Q8 C22.7C42 C22.4Q16 C23.67C23 C2×Q8⋊C4 C2×C4⋊Q8 C4⋊C4 C22×C4 C2×Q8 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 2 1 4 2 2 4 4 6 1 1

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

 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 16 0 0 0 0 1 0
,
 1 15 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 12 12 0 0 0 0 12 5
,
 7 5 0 0 0 0 7 10 0 0 0 0 0 0 10 2 0 0 0 0 9 7 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 4 9 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 11 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13

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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