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## G = (C2×Q8)⋊C8order 128 = 27

### 1st semidirect product of C2×Q8 and C8 acting via C8/C2=C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×Q8)⋊C8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — (C2×Q8)⋊C8
 Lower central C1 — C22 — C2×C4 — (C2×Q8)⋊C8
 Upper central C1 — C23 — C2×C42 — (C2×Q8)⋊C8
 Jennings C1 — C22 — C23 — C2×C42 — (C2×Q8)⋊C8

Generators and relations for (C2×Q8)⋊C8
G = < a,b,c,d | a2=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=bc >

Subgroups: 184 in 83 conjugacy classes, 34 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C23.67C23, (C2×Q8)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), SD16, Q16, C22⋊C8, C23⋊C4, C4.D4, Q8⋊C4, C4≀C2, C23⋊C8, Q8⋊C8, C23.31D4, C42.C22, C4.6Q16, (C2×Q8)⋊C8

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + - + + image C1 C2 C2 C4 C4 C8 D4 M4(2) SD16 Q16 C4≀C2 C23⋊C4 C4.D4 kernel (C2×Q8)⋊C8 C22.7C42 C23.67C23 C2×C4⋊C4 C22×Q8 C2×Q8 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 2 1 2 2 8 2 2 4 4 8 1 1

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

 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 2 0 0 0 0 16 16 0 0 0 0 0 0 0 13 0 0 0 0 13 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 13 9 0 0 0 0 0 4 0 0 0 0 0 0 1 11 0 0 0 0 6 16
,
 10 6 0 0 0 0 11 7 0 0 0 0 0 0 0 14 0 0 0 0 7 0 0 0 0 0 0 0 6 10 0 0 0 0 10 6

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,9,4,0,0,0,0,0,0,1,6,0,0,0,0,11,16],[10,11,0,0,0,0,6,7,0,0,0,0,0,0,0,7,0,0,0,0,14,0,0,0,0,0,0,0,6,10,0,0,0,0,10,6] >;

(C2×Q8)⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("(C2xQ8):C8");
// GroupNames label

G:=SmallGroup(128,4);
// by ID

G=gap.SmallGroup(128,4);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,2,56,85,232,422,387,184,794,248]);
// Polycyclic

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

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