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G = C2×He34C8order 432 = 24·33

Direct product of C2 and He34C8

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He34C8, C62.6Dic3, He39(C2×C8), (C2×He3)⋊4C8, (C6×C12).20S3, (C3×C12).66D6, (C4×He3).9C4, C4.3(He33C4), C6.5(C324C8), (C3×C12).11Dic3, (C22×He3).6C4, C12.13(C3⋊Dic3), (C4×He3).47C22, C22.2(He33C4), (C3×C6)⋊3(C3⋊C8), C324(C2×C3⋊C8), C12.90(C2×C3⋊S3), (C2×C4×He3).12C2, C6.17(C2×C3⋊Dic3), C2.1(C2×He33C4), C3.2(C2×C324C8), (C2×C12).28(C3⋊S3), (C2×He3).31(C2×C4), C4.14(C2×He3⋊C2), (C3×C6).16(C2×Dic3), (C2×C6).13(C3⋊Dic3), (C2×C4).5(He3⋊C2), SmallGroup(432,184)

Series: Derived Chief Lower central Upper central

C1C3He3 — C2×He34C8
C1C3C32He3C2×He3C4×He3He34C8 — C2×He34C8
He3 — C2×He34C8
C1C2×C12

Generators and relations for C2×He34C8
 G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 281 in 121 conjugacy classes, 59 normal (19 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C8 [×2], C2×C4, C32 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×4], C2×C8, C3×C6 [×12], C3⋊C8 [×8], C24 [×2], C2×C12, C2×C12 [×4], He3, C3×C12 [×8], C62 [×4], C2×C3⋊C8 [×4], C2×C24, C2×He3, C2×He3 [×2], C3×C3⋊C8 [×8], C6×C12 [×4], C4×He3 [×2], C22×He3, C6×C3⋊C8 [×4], He34C8 [×2], C2×C4×He3, C2×He34C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], He3⋊C2, C324C8 [×2], C2×C3⋊Dic3, He33C4 [×2], C2×He3⋊C2, C2×C324C8, He34C8 [×2], C2×He33C4, C2×He34C8

Smallest permutation representation of C2×He34C8
On 144 points
Generators in S144
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 110)(26 111)(27 112)(28 105)(29 106)(30 107)(31 108)(32 109)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 96)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 129)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 113)(121 141)(122 142)(123 143)(124 144)(125 137)(126 138)(127 139)(128 140)
(1 141 116)(2 117 142)(3 143 118)(4 119 144)(5 137 120)(6 113 138)(7 139 114)(8 115 140)(9 121 83)(10 84 122)(11 123 85)(12 86 124)(13 125 87)(14 88 126)(15 127 81)(16 82 128)(17 72 107)(18 108 65)(19 66 109)(20 110 67)(21 68 111)(22 112 69)(23 70 105)(24 106 71)(25 132 38)(26 39 133)(27 134 40)(28 33 135)(29 136 34)(30 35 129)(31 130 36)(32 37 131)(41 74 55)(42 56 75)(43 76 49)(44 50 77)(45 78 51)(46 52 79)(47 80 53)(48 54 73)(57 100 89)(58 90 101)(59 102 91)(60 92 103)(61 104 93)(62 94 97)(63 98 95)(64 96 99)
(1 49 108)(2 50 109)(3 51 110)(4 52 111)(5 53 112)(6 54 105)(7 55 106)(8 56 107)(9 96 31)(10 89 32)(11 90 25)(12 91 26)(13 92 27)(14 93 28)(15 94 29)(16 95 30)(17 115 75)(18 116 76)(19 117 77)(20 118 78)(21 119 79)(22 120 80)(23 113 73)(24 114 74)(33 88 61)(34 81 62)(35 82 63)(36 83 64)(37 84 57)(38 85 58)(39 86 59)(40 87 60)(41 71 139)(42 72 140)(43 65 141)(44 66 142)(45 67 143)(46 68 144)(47 69 137)(48 70 138)(97 136 127)(98 129 128)(99 130 121)(100 131 122)(101 132 123)(102 133 124)(103 134 125)(104 135 126)
(1 141 76)(2 77 142)(3 143 78)(4 79 144)(5 137 80)(6 73 138)(7 139 74)(8 75 140)(9 121 64)(10 57 122)(11 123 58)(12 59 124)(13 125 60)(14 61 126)(15 127 62)(16 63 128)(17 42 56)(18 49 43)(19 44 50)(20 51 45)(21 46 52)(22 53 47)(23 48 54)(24 55 41)(25 132 85)(26 86 133)(27 134 87)(28 88 135)(29 136 81)(30 82 129)(31 130 83)(32 84 131)(33 104 93)(34 94 97)(35 98 95)(36 96 99)(37 100 89)(38 90 101)(39 102 91)(40 92 103)(65 116 108)(66 109 117)(67 118 110)(68 111 119)(69 120 112)(70 105 113)(71 114 106)(72 107 115)
(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)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,110)(26,111)(27,112)(28,105)(29,106)(30,107)(31,108)(32,109)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,96)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,129)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113)(121,141)(122,142)(123,143)(124,144)(125,137)(126,138)(127,139)(128,140), (1,141,116)(2,117,142)(3,143,118)(4,119,144)(5,137,120)(6,113,138)(7,139,114)(8,115,140)(9,121,83)(10,84,122)(11,123,85)(12,86,124)(13,125,87)(14,88,126)(15,127,81)(16,82,128)(17,72,107)(18,108,65)(19,66,109)(20,110,67)(21,68,111)(22,112,69)(23,70,105)(24,106,71)(25,132,38)(26,39,133)(27,134,40)(28,33,135)(29,136,34)(30,35,129)(31,130,36)(32,37,131)(41,74,55)(42,56,75)(43,76,49)(44,50,77)(45,78,51)(46,52,79)(47,80,53)(48,54,73)(57,100,89)(58,90,101)(59,102,91)(60,92,103)(61,104,93)(62,94,97)(63,98,95)(64,96,99), (1,49,108)(2,50,109)(3,51,110)(4,52,111)(5,53,112)(6,54,105)(7,55,106)(8,56,107)(9,96,31)(10,89,32)(11,90,25)(12,91,26)(13,92,27)(14,93,28)(15,94,29)(16,95,30)(17,115,75)(18,116,76)(19,117,77)(20,118,78)(21,119,79)(22,120,80)(23,113,73)(24,114,74)(33,88,61)(34,81,62)(35,82,63)(36,83,64)(37,84,57)(38,85,58)(39,86,59)(40,87,60)(41,71,139)(42,72,140)(43,65,141)(44,66,142)(45,67,143)(46,68,144)(47,69,137)(48,70,138)(97,136,127)(98,129,128)(99,130,121)(100,131,122)(101,132,123)(102,133,124)(103,134,125)(104,135,126), (1,141,76)(2,77,142)(3,143,78)(4,79,144)(5,137,80)(6,73,138)(7,139,74)(8,75,140)(9,121,64)(10,57,122)(11,123,58)(12,59,124)(13,125,60)(14,61,126)(15,127,62)(16,63,128)(17,42,56)(18,49,43)(19,44,50)(20,51,45)(21,46,52)(22,53,47)(23,48,54)(24,55,41)(25,132,85)(26,86,133)(27,134,87)(28,88,135)(29,136,81)(30,82,129)(31,130,83)(32,84,131)(33,104,93)(34,94,97)(35,98,95)(36,96,99)(37,100,89)(38,90,101)(39,102,91)(40,92,103)(65,116,108)(66,109,117)(67,118,110)(68,111,119)(69,120,112)(70,105,113)(71,114,106)(72,107,115), (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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)>;

G:=Group( (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,110)(26,111)(27,112)(28,105)(29,106)(30,107)(31,108)(32,109)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,96)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,77)(58,78)(59,79)(60,80)(61,73)(62,74)(63,75)(64,76)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,129)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113)(121,141)(122,142)(123,143)(124,144)(125,137)(126,138)(127,139)(128,140), (1,141,116)(2,117,142)(3,143,118)(4,119,144)(5,137,120)(6,113,138)(7,139,114)(8,115,140)(9,121,83)(10,84,122)(11,123,85)(12,86,124)(13,125,87)(14,88,126)(15,127,81)(16,82,128)(17,72,107)(18,108,65)(19,66,109)(20,110,67)(21,68,111)(22,112,69)(23,70,105)(24,106,71)(25,132,38)(26,39,133)(27,134,40)(28,33,135)(29,136,34)(30,35,129)(31,130,36)(32,37,131)(41,74,55)(42,56,75)(43,76,49)(44,50,77)(45,78,51)(46,52,79)(47,80,53)(48,54,73)(57,100,89)(58,90,101)(59,102,91)(60,92,103)(61,104,93)(62,94,97)(63,98,95)(64,96,99), (1,49,108)(2,50,109)(3,51,110)(4,52,111)(5,53,112)(6,54,105)(7,55,106)(8,56,107)(9,96,31)(10,89,32)(11,90,25)(12,91,26)(13,92,27)(14,93,28)(15,94,29)(16,95,30)(17,115,75)(18,116,76)(19,117,77)(20,118,78)(21,119,79)(22,120,80)(23,113,73)(24,114,74)(33,88,61)(34,81,62)(35,82,63)(36,83,64)(37,84,57)(38,85,58)(39,86,59)(40,87,60)(41,71,139)(42,72,140)(43,65,141)(44,66,142)(45,67,143)(46,68,144)(47,69,137)(48,70,138)(97,136,127)(98,129,128)(99,130,121)(100,131,122)(101,132,123)(102,133,124)(103,134,125)(104,135,126), (1,141,76)(2,77,142)(3,143,78)(4,79,144)(5,137,80)(6,73,138)(7,139,74)(8,75,140)(9,121,64)(10,57,122)(11,123,58)(12,59,124)(13,125,60)(14,61,126)(15,127,62)(16,63,128)(17,42,56)(18,49,43)(19,44,50)(20,51,45)(21,46,52)(22,53,47)(23,48,54)(24,55,41)(25,132,85)(26,86,133)(27,134,87)(28,88,135)(29,136,81)(30,82,129)(31,130,83)(32,84,131)(33,104,93)(34,94,97)(35,98,95)(36,96,99)(37,100,89)(38,90,101)(39,102,91)(40,92,103)(65,116,108)(66,109,117)(67,118,110)(68,111,119)(69,120,112)(70,105,113)(71,114,106)(72,107,115), (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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144) );

G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,33),(24,34),(25,110),(26,111),(27,112),(28,105),(29,106),(30,107),(31,108),(32,109),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,96),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,77),(58,78),(59,79),(60,80),(61,73),(62,74),(63,75),(64,76),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,129),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,113),(121,141),(122,142),(123,143),(124,144),(125,137),(126,138),(127,139),(128,140)], [(1,141,116),(2,117,142),(3,143,118),(4,119,144),(5,137,120),(6,113,138),(7,139,114),(8,115,140),(9,121,83),(10,84,122),(11,123,85),(12,86,124),(13,125,87),(14,88,126),(15,127,81),(16,82,128),(17,72,107),(18,108,65),(19,66,109),(20,110,67),(21,68,111),(22,112,69),(23,70,105),(24,106,71),(25,132,38),(26,39,133),(27,134,40),(28,33,135),(29,136,34),(30,35,129),(31,130,36),(32,37,131),(41,74,55),(42,56,75),(43,76,49),(44,50,77),(45,78,51),(46,52,79),(47,80,53),(48,54,73),(57,100,89),(58,90,101),(59,102,91),(60,92,103),(61,104,93),(62,94,97),(63,98,95),(64,96,99)], [(1,49,108),(2,50,109),(3,51,110),(4,52,111),(5,53,112),(6,54,105),(7,55,106),(8,56,107),(9,96,31),(10,89,32),(11,90,25),(12,91,26),(13,92,27),(14,93,28),(15,94,29),(16,95,30),(17,115,75),(18,116,76),(19,117,77),(20,118,78),(21,119,79),(22,120,80),(23,113,73),(24,114,74),(33,88,61),(34,81,62),(35,82,63),(36,83,64),(37,84,57),(38,85,58),(39,86,59),(40,87,60),(41,71,139),(42,72,140),(43,65,141),(44,66,142),(45,67,143),(46,68,144),(47,69,137),(48,70,138),(97,136,127),(98,129,128),(99,130,121),(100,131,122),(101,132,123),(102,133,124),(103,134,125),(104,135,126)], [(1,141,76),(2,77,142),(3,143,78),(4,79,144),(5,137,80),(6,73,138),(7,139,74),(8,75,140),(9,121,64),(10,57,122),(11,123,58),(12,59,124),(13,125,60),(14,61,126),(15,127,62),(16,63,128),(17,42,56),(18,49,43),(19,44,50),(20,51,45),(21,46,52),(22,53,47),(23,48,54),(24,55,41),(25,132,85),(26,86,133),(27,134,87),(28,88,135),(29,136,81),(30,82,129),(31,130,83),(32,84,131),(33,104,93),(34,94,97),(35,98,95),(36,96,99),(37,100,89),(38,90,101),(39,102,91),(40,92,103),(65,116,108),(66,109,117),(67,118,110),(68,111,119),(69,120,112),(70,105,113),(71,114,106),(72,107,115)], [(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),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144)])

80 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A···6F6G···6R8A···8H12A···12H12I···12X24A···24P
order122233333344446···66···68···812···1212···1224···24
size111111666611111···16···69···91···16···69···9

80 irreducible representations

dim1111112222233333
type++++-+-
imageC1C2C2C4C4C8S3Dic3D6Dic3C3⋊C8He3⋊C2He33C4C2×He3⋊C2He33C4He34C8
kernelC2×He34C8He34C8C2×C4×He3C4×He3C22×He3C2×He3C6×C12C3×C12C3×C12C62C3×C6C2×C4C4C4C22C2
# reps121228444416444416

Matrix representation of C2×He34C8 in GL5(𝔽73)

10000
01000
007200
000720
000072
,
072000
172000
00010
00001
00100
,
10000
01000
006400
000640
000064
,
10000
01000
00080
000064
00100
,
022000
220000
000051
000510
005100

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,1,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,64,0],[0,22,0,0,0,22,0,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;

C2×He34C8 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_4C_8
% in TeX

G:=Group("C2xHe3:4C8");
// GroupNames label

G:=SmallGroup(432,184);
// by ID

G=gap.SmallGroup(432,184);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,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*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽