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## G = C2×C22.7C42order 128 = 27

### Direct product of C2 and C22.7C42

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C22.7C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C2×C22.7C42
 Lower central C1 — C2 — C2×C22.7C42
 Upper central C1 — C23×C4 — C2×C22.7C42
 Jennings C1 — C2 — C2 — C22×C4 — C2×C22.7C42

Generators and relations for C2×C22.7C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 356 in 264 conjugacy classes, 172 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22 [×3], C22 [×32], C8 [×8], C2×C4 [×2], C2×C4 [×34], C2×C4 [×24], C23, C23 [×14], C42 [×8], C2×C8 [×8], C2×C8 [×24], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C2×C42 [×4], C2×C42 [×4], C22×C8 [×12], C22×C8 [×8], C23×C4, C23×C4 [×2], C22.7C42 [×4], C22×C42, C23×C8 [×2], C2×C22.7C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C2×C8 [×12], M4(2) [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C22×C8 [×2], C2×M4(2) [×2], C22.7C42 [×8], C2×C2.C42, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C2×C22.7C42

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

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

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

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

80 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 4Q ··· 4AF 8A ··· 8AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 M4(2) kernel C2×C22.7C42 C22.7C42 C22×C42 C23×C8 C2×C42 C22×C8 C23×C4 C22×C4 C22×C4 C22×C4 C23 # reps 1 4 1 2 4 16 4 32 6 2 8

Matrix representation of C2×C22.7C42 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 2 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 14 8 0 0 0 3 3
,
 1 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 16 15 0 0 0 0 1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,14,3,0,0,0,8,3],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,15,1] >;

C2×C22.7C42 in GAP, Magma, Sage, TeX

C_2\times C_2^2._7C_4^2
% in TeX

G:=Group("C2xC2^2.7C4^2");
// GroupNames label

G:=SmallGroup(128,459);
// by ID

G=gap.SmallGroup(128,459);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,172]);
// Polycyclic

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

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