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G = C2×C428C4order 128 = 27

Direct product of C2 and C428C4

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

Aliases: C2×C428C4, C23.163C24, C24.641C23, C4244(C2×C4), (C2×C42)⋊20C4, (C22×C4).93Q8, C23.822(C2×D4), (C22×C4).593D4, C23.139(C2×Q8), C22.54(C23×C4), (C23×C4).33C22, (C22×C42).18C2, C22.63(C22×D4), C23.355(C4○D4), C22.18(C22×Q8), C23.280(C22×C4), (C22×C4).441C23, (C2×C42).1085C22, C22.72(C4.4D4), C22.24(C42.C2), C22.65(C42⋊C2), C2.C42.462C22, C4.56(C2×C4⋊C4), C2.5(C22×C4⋊C4), (C2×C4).822(C2×D4), C22.70(C2×C4⋊C4), (C2×C4).223(C2×Q8), C2.1(C2×C4.4D4), (C2×C4).146(C4⋊C4), C2.1(C2×C42.C2), (C22×C4⋊C4).19C2, C2.9(C2×C42⋊C2), C22.56(C2×C4○D4), (C2×C4⋊C4).783C22, (C2×C4).485(C22×C4), (C22×C4).451(C2×C4), (C2×C2.C42).7C2, SmallGroup(128,1013)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C428C4
C1C2C22C23C24C23×C4C22×C42 — C2×C428C4
C1C22 — C2×C428C4
C1C24 — C2×C428C4
C1C23 — C2×C428C4

Generators and relations for C2×C428C4
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 540 in 356 conjugacy classes, 220 normal (10 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×16], C22 [×3], C22 [×32], C2×C4 [×36], C2×C4 [×64], C23, C23 [×14], C42 [×16], C4⋊C4 [×16], C22×C4 [×34], C22×C4 [×32], C24, C2.C42 [×16], C2×C42 [×12], C2×C4⋊C4 [×8], C2×C4⋊C4 [×8], C23×C4, C23×C4 [×6], C2×C2.C42 [×4], C428C4 [×8], C22×C42, C22×C4⋊C4 [×2], C2×C428C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×8], C24, C2×C4⋊C4 [×12], C42⋊C2 [×8], C4.4D4 [×8], C42.C2 [×8], C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×4], C428C4 [×8], C22×C4⋊C4, C2×C42⋊C2 [×2], C2×C4.4D4 [×2], C2×C42.C2 [×2], C2×C428C4

Smallest permutation representation of C2×C428C4
Regular action on 128 points
Generators in S128
(1 8)(2 5)(3 6)(4 7)(9 31)(10 32)(11 29)(12 30)(13 50)(14 51)(15 52)(16 49)(17 35)(18 36)(19 33)(20 34)(21 39)(22 40)(23 37)(24 38)(25 61)(26 62)(27 63)(28 64)(41 93)(42 94)(43 95)(44 96)(45 79)(46 80)(47 77)(48 78)(53 59)(54 60)(55 57)(56 58)(65 84)(66 81)(67 82)(68 83)(69 88)(70 85)(71 86)(72 87)(73 92)(74 89)(75 90)(76 91)(97 116)(98 113)(99 114)(100 115)(101 120)(102 117)(103 118)(104 119)(105 124)(106 121)(107 122)(108 123)(109 128)(110 125)(111 126)(112 127)
(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 25 18 46)(2 26 19 47)(3 27 20 48)(4 28 17 45)(5 62 33 77)(6 63 34 78)(7 64 35 79)(8 61 36 80)(9 21 15 58)(10 22 16 59)(11 23 13 60)(12 24 14 57)(29 37 50 54)(30 38 51 55)(31 39 52 56)(32 40 49 53)(41 102 125 69)(42 103 126 70)(43 104 127 71)(44 101 128 72)(65 113 106 92)(66 114 107 89)(67 115 108 90)(68 116 105 91)(73 84 98 121)(74 81 99 122)(75 82 100 123)(76 83 97 124)(85 94 118 111)(86 95 119 112)(87 96 120 109)(88 93 117 110)
(1 102 10 90)(2 70 11 116)(3 104 12 92)(4 72 9 114)(5 85 29 97)(6 119 30 73)(7 87 31 99)(8 117 32 75)(13 91 19 103)(14 113 20 71)(15 89 17 101)(16 115 18 69)(21 105 28 42)(22 65 25 127)(23 107 26 44)(24 67 27 125)(33 118 50 76)(34 86 51 98)(35 120 52 74)(36 88 49 100)(37 122 62 96)(38 82 63 110)(39 124 64 94)(40 84 61 112)(41 57 108 48)(43 59 106 46)(45 126 58 68)(47 128 60 66)(53 121 80 95)(54 81 77 109)(55 123 78 93)(56 83 79 111)

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

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

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

56 conjugacy classes

class 1 2A···2O4A···4X4Y···4AN
order12···24···44···4
size11···12···24···4

56 irreducible representations

dim111111222
type++++++-
imageC1C2C2C2C2C4D4Q8C4○D4
kernelC2×C428C4C2×C2.C42C428C4C22×C42C22×C4⋊C4C2×C42C22×C4C22×C4C23
# reps14812164416

Matrix representation of C2×C428C4 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
010000
003000
000200
000033
000042
,
400000
040000
003000
000200
000020
000002
,
200000
010000
000100
004000
000044
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,4,0,0,0,0,3,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1] >;

C2×C428C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_8C_4
% in TeX

G:=Group("C2xC4^2:8C4");
// GroupNames label

G:=SmallGroup(128,1013);
// by ID

G=gap.SmallGroup(128,1013);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100]);
// Polycyclic

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

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