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## G = C2×C42.30C22order 128 = 27

### Direct product of C2 and C42.30C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.30C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2×C42.30C22
 Lower central C1 — C2 — C2×C4 — C2×C42.30C22
 Upper central C1 — C23 — C2×C42 — C2×C42.30C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.30C22

Generators and relations for C2×C42.30C22
G = < a,b,c,d,e | a2=b4=c4=1, d2=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, ebe-1=bc2, dcd-1=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 324 in 192 conjugacy classes, 100 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C8⋊C4, Q8⋊C4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C22×Q8, C2×C8⋊C4, C2×Q8⋊C4, C42.30C22, C2×C42.C2, C2×C4⋊Q8, C2×C42.30C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8.C22, C22×D4, C2×C4○D4, C42.30C22, C2×C4.4D4, C2×C8.C22, C2×C42.30C22

Smallest permutation representation of C2×C42.30C22
Regular action on 128 points
Generators in S128
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 78)(10 79)(11 80)(12 73)(13 74)(14 75)(15 76)(16 77)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)(41 95)(42 96)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(97 116)(98 117)(99 118)(100 119)(101 120)(102 113)(103 114)(104 115)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(1 60 21 11)(2 57 22 16)(3 62 23 13)(4 59 24 10)(5 64 17 15)(6 61 18 12)(7 58 19 9)(8 63 20 14)(25 74 55 36)(26 79 56 33)(27 76 49 38)(28 73 50 35)(29 78 51 40)(30 75 52 37)(31 80 53 34)(32 77 54 39)(41 113 107 66)(42 118 108 71)(43 115 109 68)(44 120 110 65)(45 117 111 70)(46 114 112 67)(47 119 105 72)(48 116 106 69)(81 91 98 122)(82 96 99 127)(83 93 100 124)(84 90 101 121)(85 95 102 126)(86 92 103 123)(87 89 104 128)(88 94 97 125)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 115 5 119)(2 71 6 67)(3 113 7 117)(4 69 8 65)(9 111 13 107)(10 48 14 44)(11 109 15 105)(12 46 16 42)(17 72 21 68)(18 114 22 118)(19 70 23 66)(20 120 24 116)(25 85 29 81)(26 97 30 101)(27 83 31 87)(28 103 32 99)(33 125 37 121)(34 89 38 93)(35 123 39 127)(36 95 40 91)(41 58 45 62)(43 64 47 60)(49 100 53 104)(50 86 54 82)(51 98 55 102)(52 84 56 88)(57 108 61 112)(59 106 63 110)(73 92 77 96)(74 126 78 122)(75 90 79 94)(76 124 80 128)
(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)

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

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

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

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 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 C8.C22 kernel C2×C42.30C22 C2×C8⋊C4 C2×Q8⋊C4 C42.30C22 C2×C42.C2 C2×C4⋊Q8 C42 C22×C4 C2×C4 C22 # reps 1 1 4 8 1 1 2 2 8 4

Matrix representation of C2×C42.30C22 in GL8(𝔽17)

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

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

C2×C42.30C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{30}C_2^2
% in TeX

G:=Group("C2xC4^2.30C2^2");
// GroupNames label

G:=SmallGroup(128,1866);
// by ID

G=gap.SmallGroup(128,1866);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,680,758,723,100,2804,172,4037,124]);
// Polycyclic

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

׿
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