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## G = (C2×Q8).8Q8order 128 = 27

### 8th non-split extension by C2×Q8 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×Q8).8Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×Q8⋊C4 — (C2×Q8).8Q8
 Lower central C1 — C2 — C22×C4 — (C2×Q8).8Q8
 Upper central C1 — C23 — C2×C42 — (C2×Q8).8Q8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×Q8).8Q8

Generators and relations for (C2×Q8).8Q8
G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=b2cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=abc, ebe-1=b-1c, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 248 in 124 conjugacy classes, 52 normal (44 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×Q8, C2.C42, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4, C2×C4⋊C8, C2×C4.Q8, (C2×Q8).8Q8
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C422C2, C2×SD16, C4○D8, C8.C22, C23.Q8, D4.D4, Q8.D4, C88D4, C8.D4, Q8⋊Q8, Q8.Q8, (C2×Q8).8Q8

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

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

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

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

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 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8 SD16 C4○D4 C4○D8 C8.C22 kernel (C2×Q8).8Q8 C22.4Q16 C23.65C23 C23.67C23 C2×Q8⋊C4 C2×C4⋊C8 C2×C4.Q8 C4⋊C4 C2×C8 C22×C4 C2×Q8 C2×C4 C2×C4 C22 C22 # reps 1 1 1 1 2 1 1 2 2 2 2 4 6 4 2

Matrix representation of (C2×Q8).8Q8 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
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 4 0 0 0 0 4 0
,
 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 15 0 0 0 0 1 1 0 0 0 0 0 0 5 5 0 0 0 0 12 5
,
 10 4 0 0 0 0 13 7 0 0 0 0 0 0 8 2 0 0 0 0 10 9 0 0 0 0 0 0 14 14 0 0 0 0 14 3

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],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;

(C2×Q8).8Q8 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._8Q_8
% in TeX

G:=Group("(C2xQ8).8Q8");
// GroupNames label

G:=SmallGroup(128,798);
// by ID

G=gap.SmallGroup(128,798);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^4=b^2,e^2=b^2*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=a*b*c,e*b*e^-1=b^-1*c,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
// generators/relations

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