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## G = C2×C4.10D8order 128 = 27

### Direct product of C2 and C4.10D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4.10D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊Q8 — C2×C4.10D8
 Lower central C1 — C22 — C2×C4 — C2×C4.10D8
 Upper central C1 — C23 — C2×C42 — C2×C4.10D8
 Jennings C1 — C22 — C22 — C42 — C2×C4.10D8

Generators and relations for C2×C4.10D8
G = < a,b,c,d | a2=b4=c8=1, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=bc-1 >

Subgroups: 244 in 128 conjugacy classes, 68 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×6], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×8], C2×C4 [×10], Q8 [×8], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×8], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C4.10D8 [×4], C2×C4⋊C8 [×2], C2×C4⋊Q8, C2×C4.10D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C2×D4 [×2], C4.10D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16 [×2], C2×Q16, C4.10D8 [×4], C2×C4.10D4, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.10D8

Smallest permutation representation of C2×C4.10D8
Regular action on 128 points
Generators in S128
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)(73 109)(74 110)(75 111)(76 112)(77 105)(78 106)(79 107)(80 108)(113 125)(114 126)(115 127)(116 128)(117 121)(118 122)(119 123)(120 124)
(1 33 19 93)(2 94 20 34)(3 35 21 95)(4 96 22 36)(5 37 23 89)(6 90 24 38)(7 39 17 91)(8 92 18 40)(9 83 125 111)(10 112 126 84)(11 85 127 105)(12 106 128 86)(13 87 121 107)(14 108 122 88)(15 81 123 109)(16 110 124 82)(25 66 101 63)(26 64 102 67)(27 68 103 57)(28 58 104 69)(29 70 97 59)(30 60 98 71)(31 72 99 61)(32 62 100 65)(41 115 77 55)(42 56 78 116)(43 117 79 49)(44 50 80 118)(45 119 73 51)(46 52 74 120)(47 113 75 53)(48 54 76 114)
(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 76 93 54 19 48 33 114)(2 53 34 75 20 113 94 47)(3 74 95 52 21 46 35 120)(4 51 36 73 22 119 96 45)(5 80 89 50 23 44 37 118)(6 49 38 79 24 117 90 43)(7 78 91 56 17 42 39 116)(8 55 40 77 18 115 92 41)(9 62 111 32 125 65 83 100)(10 31 84 61 126 99 112 72)(11 60 105 30 127 71 85 98)(12 29 86 59 128 97 106 70)(13 58 107 28 121 69 87 104)(14 27 88 57 122 103 108 68)(15 64 109 26 123 67 81 102)(16 25 82 63 124 101 110 66)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - - image C1 C2 C2 C2 C4 C4 D4 D4 D8 SD16 Q16 C4.10D4 kernel C2×C4.10D8 C4.10D8 C2×C4⋊C8 C2×C4⋊Q8 C2×C4⋊C4 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 2 1 4 4 2 2 4 8 4 2

Matrix representation of C2×C4.10D8 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
,
 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 0 16 0 0 0 0 1 0
,
 3 3 0 0 0 0 14 3 0 0 0 0 0 0 14 14 0 0 0 0 3 14 0 0 0 0 0 0 11 4 0 0 0 0 4 6
,
 4 11 0 0 0 0 11 13 0 0 0 0 0 0 4 11 0 0 0 0 11 13 0 0 0 0 0 0 12 5 0 0 0 0 12 12

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,11,4,0,0,0,0,4,6],[4,11,0,0,0,0,11,13,0,0,0,0,0,0,4,11,0,0,0,0,11,13,0,0,0,0,0,0,12,12,0,0,0,0,5,12] >;

C2×C4.10D8 in GAP, Magma, Sage, TeX

C_2\times C_4._{10}D_8
% in TeX

G:=Group("C2xC4.10D8");
// GroupNames label

G:=SmallGroup(128,271);
// by ID

G=gap.SmallGroup(128,271);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971,242]);
// Polycyclic

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

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