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## G = (C2×C8).1Q8order 128 = 27

### 1st non-split extension by C2×C8 of 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).1Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C2.D8 — (C2×C8).1Q8
 Lower central C1 — C2 — C22×C4 — (C2×C8).1Q8
 Upper central C1 — C23 — C2×C42 — (C2×C8).1Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).1Q8

Generators and relations for (C2×C8).1Q8
G = < a,b,c,d | a2=b8=c4=1, d2=ac2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 232 in 115 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22.7C42, C22.4Q16 [×2], C429C4, C23.65C23, C2×C2.D8 [×2], (C2×C8).1Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D8 [×2], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.81C23, C4⋊D8, C42Q16, C22.D8, C23.48D4, C82Q8, C8⋊Q8, (C2×C8).1Q8

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

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

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

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

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 Q8 D4 D8 Q16 C4○D4 C8⋊C22 C8.C22 kernel (C2×C8).1Q8 C22.7C42 C22.4Q16 C42⋊9C4 C23.65C23 C2×C2.D8 C4⋊C4 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 1 1 2 2 4 2 4 4 6 1 1

Matrix representation of (C2×C8).1Q8 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 16 0 0 0 0 0 0 16
,
 10 9 0 0 0 0 6 7 0 0 0 0 0 0 15 0 0 0 0 0 0 8 0 0 0 0 0 0 0 4 0 0 0 0 4 0
,
 6 12 0 0 0 0 7 11 0 0 0 0 0 0 0 15 0 0 0 0 9 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 6 2 0 0 0 0 7 11 0 0 0 0 0 0 0 9 0 0 0 0 15 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,6,0,0,0,0,9,7,0,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[6,7,0,0,0,0,12,11,0,0,0,0,0,0,0,9,0,0,0,0,15,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[6,7,0,0,0,0,2,11,0,0,0,0,0,0,0,15,0,0,0,0,9,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

(C2×C8).1Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8)._1Q_8
% in TeX

G:=Group("(C2xC8).1Q8");
// GroupNames label

G:=SmallGroup(128,815);
// by ID

G=gap.SmallGroup(128,815);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,2028,1027,124]);
// Polycyclic

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

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