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## G = C2.(C8⋊D4)  order 128 = 27

### 4th central extension by C2 of C8⋊D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C8⋊D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2.(C8⋊D4)
 Lower central C1 — C2 — C2×C4 — C2.(C8⋊D4)
 Upper central C1 — C23 — C2×C42 — C2.(C8⋊D4)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C8⋊D4)

Generators and relations for C2.(C8⋊D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab3, dcd-1=c-1 >

Subgroups: 244 in 124 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, 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, C8⋊C4, Q8⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×C8⋊C4, C2×Q8⋊C4, C2.(C8⋊D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C422C2, C8⋊C22, C8.C22, C24.C22, SD16⋊C4, Q16⋊C4, C8⋊D4, C8.D4, C42.28C22, C42.30C22, C2.(C8⋊D4)

Smallest permutation representation of C2.(C8⋊D4)
Regular action on 128 points
Generators in S128
(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(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)
(1 33 26 118)(2 15 27 89)(3 39 28 116)(4 13 29 95)(5 37 30 114)(6 11 31 93)(7 35 32 120)(8 9 25 91)(10 51 92 69)(12 49 94 67)(14 55 96 65)(16 53 90 71)(17 82 73 125)(18 105 74 48)(19 88 75 123)(20 111 76 46)(21 86 77 121)(22 109 78 44)(23 84 79 127)(24 107 80 42)(34 52 119 70)(36 50 113 68)(38 56 115 66)(40 54 117 72)(41 58 106 97)(43 64 108 103)(45 62 110 101)(47 60 112 99)(57 83 104 126)(59 81 98 124)(61 87 100 122)(63 85 102 128)
(1 47 71 123)(2 126 72 42)(3 45 65 121)(4 124 66 48)(5 43 67 127)(6 122 68 46)(7 41 69 125)(8 128 70 44)(9 102 34 78)(10 73 35 97)(11 100 36 76)(12 79 37 103)(13 98 38 74)(14 77 39 101)(15 104 40 80)(16 75 33 99)(17 120 58 92)(18 95 59 115)(19 118 60 90)(20 93 61 113)(21 116 62 96)(22 91 63 119)(23 114 64 94)(24 89 57 117)(25 85 52 109)(26 112 53 88)(27 83 54 107)(28 110 55 86)(29 81 56 105)(30 108 49 84)(31 87 50 111)(32 106 51 82)

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

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

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

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 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C2.(C8⋊D4) C22.4Q16 C23.65C23 C23.67C23 C2×C8⋊C4 C2×Q8⋊C4 Q8⋊C4 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 2 1 1 1 2 8 2 2 8 1 3

Matrix representation of C2.(C8⋊D4) in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 2 0 0 0 0 3 3 3 9 0 0 0 0 7 12 1 1 0 0 0 0 10 1 10 13
,
 13 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 13 7 0 0 0 0 0 0 0 0 8 15 14 5 0 0 0 0 4 3 13 3 0 0 0 0 6 10 12 7 0 0 0 0 16 7 7 11
,
 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 13 7 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 12 16 13 0 0 0 0 14 5 10 1 0 0 0 0 6 6 7 9 0 0 0 0 2 1 13 12

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,7,10,0,0,0,0,16,3,12,1,0,0,0,0,0,3,1,10,0,0,0,0,2,9,1,13],[13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,10,13,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,8,4,6,16,0,0,0,0,15,3,10,7,0,0,0,0,14,13,12,7,0,0,0,0,5,3,7,11],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,13,10,0,0,0,0,0,0,7,4,0,0,0,0,0,0,0,0,10,14,6,2,0,0,0,0,12,5,6,1,0,0,0,0,16,10,7,13,0,0,0,0,13,1,9,12] >;

C2.(C8⋊D4) in GAP, Magma, Sage, TeX

C_2.(C_8\rtimes D_4)
% in TeX

G:=Group("C2.(C8:D4)");
// GroupNames label

G:=SmallGroup(128,667);
// by ID

G=gap.SmallGroup(128,667);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,2019,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^3,d*c*d^-1=c^-1>;
// generators/relations

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