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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8)⋊Q8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.65C23 — (C2×C8)⋊Q8
 Lower central C1 — C2 — C22×C4 — (C2×C8)⋊Q8
 Upper central C1 — C23 — C2×C42 — (C2×C8)⋊Q8
 Jennings C1 — C2 — C2 — C22×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, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2.C42, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C23.65C23, C2×C4.Q8, C2×C4⋊Q8, (C2×C8)⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C22≀C2, C22⋊Q8, C4⋊Q8, C2×SD16, 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 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(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)(73 102)(74 103)(75 104)(76 97)(77 98)(78 99)(79 100)(80 101)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)
(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 47 84 77)(34 42 85 80)(35 45 86 75)(36 48 87 78)(37 43 88 73)(38 46 81 76)(39 41 82 79)(40 44 83 74)(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 75 54 45)(3 97 55 22)(4 77 56 47)(5 99 49 24)(6 79 50 41)(7 101 51 18)(8 73 52 43)(9 84 107 33)(10 92 108 25)(11 86 109 35)(12 94 110 27)(13 88 111 37)(14 96 112 29)(15 82 105 39)(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 62 85 71)(36 64 87 65)(38 58 81 67)(40 60 83 69)(42 115 80 125)(44 117 74 127)(46 119 76 121)(48 113 78 123)

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

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

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

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 Q8 Q8 D4 SD16 C4○D4 C8⋊C22 C8.C22 kernel (C2×C8)⋊Q8 C22.7C42 C22.4Q16 C23.65C23 C2×C4.Q8 C2×C4⋊Q8 C4⋊C4 C4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 2 1 4 2 4 2 8 2 1 1

Matrix representation of (C2×C8)⋊Q8 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
,
 10 7 0 0 0 0 5 0 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 0 0 0 0 10 4 0 0 0 0 13 7
,
 4 0 0 0 0 0 4 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 4 10 0 0 0 0 7 13
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 13 7 0 0 0 0 10 4

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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