Copied to
clipboard

G = C2×C425C4order 128 = 27

Direct product of C2 and C425C4

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

Aliases: C2×C425C4, C24.642C23, C23.164C24, (C2×C42)⋊15C4, C4241(C2×C4), C22.55(C23×C4), (C22×C42).10C2, C23.356(C4○D4), (C22×C4).442C23, (C23×C4).644C22, C23.281(C22×C4), (C2×C42).1000C22, C22.66(C42⋊C2), C22.29(C422C2), C2.C42.463C22, C2.1(C2×C422C2), C22.57(C2×C4○D4), (C2×C4).486(C22×C4), (C22×C4).452(C2×C4), C2.10(C2×C42⋊C2), (C2×C2.C42).8C2, SmallGroup(128,1014)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C425C4
C1C2C22C23C24C23×C4C22×C42 — C2×C425C4
C1C22 — C2×C425C4
C1C24 — C2×C425C4
C1C23 — C2×C425C4

Generators and relations for C2×C425C4
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >

Subgroups: 492 in 312 conjugacy classes, 180 normal (6 characteristic)
C1, C2 [×15], C4 [×20], C22, C22 [×34], C2×C4 [×12], C2×C4 [×76], C23, C23 [×14], C42 [×16], C22×C4 [×26], C22×C4 [×36], C24, C2.C42 [×24], C2×C42 [×12], C23×C4 [×7], C2×C2.C42 [×6], C425C4 [×8], C22×C42, C2×C425C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×12], C24, C42⋊C2 [×12], C422C2 [×16], C23×C4, C2×C4○D4 [×6], C425C4 [×8], C2×C42⋊C2 [×3], C2×C422C2 [×4], C2×C425C4

Smallest permutation representation of C2×C425C4
Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 13)(4 14)(5 111)(6 112)(7 109)(8 110)(9 20)(10 17)(11 18)(12 19)(21 98)(22 99)(23 100)(24 97)(25 80)(26 77)(27 78)(28 79)(29 76)(30 73)(31 74)(32 75)(33 60)(34 57)(35 58)(36 59)(37 64)(38 61)(39 62)(40 63)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)(49 53)(50 54)(51 55)(52 56)(65 86)(66 87)(67 88)(68 85)(69 82)(70 83)(71 84)(72 81)(101 122)(102 123)(103 124)(104 121)(105 118)(106 119)(107 120)(108 117)(113 126)(114 127)(115 128)(116 125)
(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 27 20 29)(2 28 17 30)(3 25 18 31)(4 26 19 32)(5 105 126 103)(6 106 127 104)(7 107 128 101)(8 108 125 102)(9 76 15 78)(10 73 16 79)(11 74 13 80)(12 75 14 77)(21 60 37 54)(22 57 38 55)(23 58 39 56)(24 59 40 53)(33 64 50 98)(34 61 51 99)(35 62 52 100)(36 63 49 97)(41 69 47 67)(42 70 48 68)(43 71 45 65)(44 72 46 66)(81 96 87 90)(82 93 88 91)(83 94 85 92)(84 95 86 89)(109 120 115 122)(110 117 116 123)(111 118 113 124)(112 119 114 121)
(1 65 36 119)(2 72 33 122)(3 67 34 117)(4 70 35 124)(5 73 94 21)(6 80 95 38)(7 75 96 23)(8 78 93 40)(9 84 53 104)(10 87 54 107)(11 82 55 102)(12 85 56 105)(13 88 57 108)(14 83 58 103)(15 86 59 106)(16 81 60 101)(17 66 50 120)(18 69 51 123)(19 68 52 118)(20 71 49 121)(22 127 74 89)(24 125 76 91)(25 45 61 112)(26 44 62 115)(27 47 63 110)(28 42 64 113)(29 41 97 116)(30 48 98 111)(31 43 99 114)(32 46 100 109)(37 126 79 92)(39 128 77 90)

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim111112
type++++
imageC1C2C2C2C4C4○D4
kernelC2×C425C4C2×C2.C42C425C4C22×C42C2×C42C23
# reps16811624

Matrix representation of C2×C425C4 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
010000
000200
003000
000042
000041
,
400000
010000
000400
001000
000030
000003
,
200000
010000
000300
003000
000042
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],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;

C2×C425C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_5C_4
% in TeX

G:=Group("C2xC4^2:5C4");
// GroupNames label

G:=SmallGroup(128,1014);
// by ID

G=gap.SmallGroup(128,1014);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,100]);
// 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^-1>;
// generators/relations

׿
×
𝔽