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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 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8, (C2×Q8).8Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×SD16, C4○D8, C8.C22 [×2], 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 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 125)(34 126)(35 127)(36 128)(37 121)(38 122)(39 123)(40 124)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(65 102)(66 103)(67 104)(68 97)(69 98)(70 99)(71 100)(72 101)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)
(1 71 5 67)(2 30 6 26)(3 69 7 65)(4 28 8 32)(9 35 13 39)(10 112 14 108)(11 33 15 37)(12 110 16 106)(17 29 21 25)(18 68 22 72)(19 27 23 31)(20 66 24 70)(34 46 38 42)(36 44 40 48)(41 105 45 109)(43 111 47 107)(49 59 53 63)(50 74 54 78)(51 57 55 61)(52 80 56 76)(58 101 62 97)(60 99 64 103)(73 102 77 98)(75 100 79 104)(81 94 85 90)(82 123 86 127)(83 92 87 96)(84 121 88 125)(89 116 93 120)(91 114 95 118)(113 126 117 122)(115 124 119 128)
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 118 13 114)(10 119 14 115)(11 120 15 116)(12 113 16 117)(17 75 21 79)(18 76 22 80)(19 77 23 73)(20 78 24 74)(25 100 29 104)(26 101 30 97)(27 102 31 98)(28 103 32 99)(33 93 37 89)(34 94 38 90)(35 95 39 91)(36 96 40 92)(41 84 45 88)(42 85 46 81)(43 86 47 82)(44 87 48 83)(49 65 53 69)(50 66 54 70)(51 67 55 71)(52 68 56 72)(105 125 109 121)(106 126 110 122)(107 127 111 123)(108 128 112 124)
(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)
(1 109 63 127)(2 112 64 122)(3 107 57 125)(4 110 58 128)(5 105 59 123)(6 108 60 126)(7 111 61 121)(8 106 62 124)(9 29 116 102)(10 32 117 97)(11 27 118 100)(12 30 119 103)(13 25 120 98)(14 28 113 101)(15 31 114 104)(16 26 115 99)(17 37 73 95)(18 40 74 90)(19 35 75 93)(20 38 76 96)(21 33 77 91)(22 36 78 94)(23 39 79 89)(24 34 80 92)(41 69 82 55)(42 72 83 50)(43 67 84 53)(44 70 85 56)(45 65 86 51)(46 68 87 54)(47 71 88 49)(48 66 81 52)

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

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

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

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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