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G = C2×He3⋊3C8  order 432 = 24·33

Direct product of C2 and He3⋊3C8

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3⋊3C8, C62.4C12, C62.1Dic3, (C3×C6)⋊2C24, He3⋊8(C2×C8), (C6×C12).8C6, (C2×He3)⋊3C8, C12.92(S3×C6), (C6×C12).16S3, (C3×C12).3C12, C32⋊3(C2×C24), C32⋊4C8⋊7C6, (C3×C12).60D6, (C4×He3).8C4, (C3×C12).8Dic3, C6.10(C6×Dic3), C4.3(C32⋊C12), C12.13(C3×Dic3), (C22×He3).4C4, (C4×He3).43C22, C22.2(C32⋊C12), C6.5(C3×C3⋊C8), C3.2(C6×C3⋊C8), (C3×C6)⋊2(C3⋊C8), C32⋊3(C2×C3⋊C8), (C2×C32⋊4C8)⋊C3, (C3×C6).5(C2×C12), (C2×C4×He3).10C2, (C3×C12).15(C2×C6), (C2×C12).31(C3×S3), C2.1(C2×C32⋊C12), C4.14(C2×C32⋊C6), (C3×C6).6(C2×Dic3), (C2×C4).5(C32⋊C6), (C2×He3).26(C2×C4), (C2×C6).14(C3×Dic3), SmallGroup(432,136)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×He3⋊3C8
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3C8 — C2×He3⋊3C8
Lower central
C32 — C2×He3⋊3C8
Upper central
C1 — C2×C4

Generators and relations for C2×He3⋊3C8
 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, ece-1=c-1, de=ed >

Subgroups: 257 in 93 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C12, C12, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, He3, C3×C12, C3×C12, C62, C62, C2×C3⋊C8, C2×C24, C2×He3, C2×He3, C3×C3⋊C8, C32⋊4C8, C6×C12, C6×C12, C4×He3, C22×He3, C6×C3⋊C8, C2×C32⋊4C8, He3⋊3C8, C2×C4×He3, C2×He3⋊3C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, C32⋊C6, C3×C3⋊C8, C6×Dic3, C32⋊C12, C2×C32⋊C6, C6×C3⋊C8, He3⋊3C8, C2×C32⋊C12, C2×He3⋊3C8

Smallest permutation representation of C2×He3⋊3C8
►On 144 points
Generators in S144
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(41 126)(42 127)(43 128)(44 121)(45 122)(46 123)(47 124)(48 125)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 105)(72 106)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 137)(97 120)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)

(1 39 143)(2 144 40)(3 33 137)(4 138 34)(5 35 139)(6 140 36)(7 37 141)(8 142 38)(17 96 49)(18 50 89)(19 90 51)(20 52 91)(21 92 53)(22 54 93)(23 94 55)(24 56 95)(25 97 67)(26 68 98)(27 99 69)(28 70 100)(29 101 71)(30 72 102)(31 103 65)(32 66 104)(81 106 117)(82 118 107)(83 108 119)(84 120 109)(85 110 113)(86 114 111)(87 112 115)(88 116 105)

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

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

(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)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(41,126)(42,127)(43,128)(44,121)(45,122)(46,123)(47,124)(48,125)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,105)(72,106)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,137)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,97,67)(26,68,98)(27,99,69)(28,70,100)(29,101,71)(30,72,102)(31,103,65)(32,66,104)(81,106,117)(82,118,107)(83,108,119)(84,120,109)(85,110,113)(86,114,111)(87,112,115)(88,116,105), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(9,74,44)(10,45,75)(11,76,46)(12,47,77)(13,78,48)(14,41,79)(15,80,42)(16,43,73)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,67,97)(26,98,68)(27,69,99)(28,100,70)(29,71,101)(30,102,72)(31,65,103)(32,104,66)(57,126,135)(58,136,127)(59,128,129)(60,130,121)(61,122,131)(62,132,123)(63,124,133)(64,134,125)(81,117,106)(82,107,118)(83,119,108)(84,109,120)(85,113,110)(86,111,114)(87,115,112)(88,105,116), (1,48,106)(2,41,107)(3,42,108)(4,43,109)(5,44,110)(6,45,111)(7,46,112)(8,47,105)(9,85,35)(10,86,36)(11,87,37)(12,88,38)(13,81,39)(14,82,40)(15,83,33)(16,84,34)(17,58,32)(18,59,25)(19,60,26)(20,61,27)(21,62,28)(22,63,29)(23,64,30)(24,57,31)(49,127,66)(50,128,67)(51,121,68)(52,122,69)(53,123,70)(54,124,71)(55,125,72)(56,126,65)(73,120,138)(74,113,139)(75,114,140)(76,115,141)(77,116,142)(78,117,143)(79,118,144)(80,119,137)(89,129,97)(90,130,98)(91,131,99)(92,132,100)(93,133,101)(94,134,102)(95,135,103)(96,136,104), (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( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(41,126)(42,127)(43,128)(44,121)(45,122)(46,123)(47,124)(48,125)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,105)(72,106)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,137)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,97,67)(26,68,98)(27,99,69)(28,70,100)(29,101,71)(30,72,102)(31,103,65)(32,66,104)(81,106,117)(82,118,107)(83,108,119)(84,120,109)(85,110,113)(86,114,111)(87,112,115)(88,116,105), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(9,74,44)(10,45,75)(11,76,46)(12,47,77)(13,78,48)(14,41,79)(15,80,42)(16,43,73)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,67,97)(26,98,68)(27,69,99)(28,100,70)(29,71,101)(30,102,72)(31,65,103)(32,104,66)(57,126,135)(58,136,127)(59,128,129)(60,130,121)(61,122,131)(62,132,123)(63,124,133)(64,134,125)(81,117,106)(82,107,118)(83,119,108)(84,109,120)(85,113,110)(86,111,114)(87,115,112)(88,105,116), (1,48,106)(2,41,107)(3,42,108)(4,43,109)(5,44,110)(6,45,111)(7,46,112)(8,47,105)(9,85,35)(10,86,36)(11,87,37)(12,88,38)(13,81,39)(14,82,40)(15,83,33)(16,84,34)(17,58,32)(18,59,25)(19,60,26)(20,61,27)(21,62,28)(22,63,29)(23,64,30)(24,57,31)(49,127,66)(50,128,67)(51,121,68)(52,122,69)(53,123,70)(54,124,71)(55,125,72)(56,126,65)(73,120,138)(74,113,139)(75,114,140)(76,115,141)(77,116,142)(78,117,143)(79,118,144)(80,119,137)(89,129,97)(90,130,98)(91,131,99)(92,132,100)(93,133,101)(94,134,102)(95,135,103)(96,136,104), (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([[(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,84),(26,85),(27,86),(28,87),(29,88),(30,81),(31,82),(32,83),(41,126),(42,127),(43,128),(44,121),(45,122),(46,123),(47,124),(48,125),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,105),(72,106),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,137),(97,120),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119)], [(1,39,143),(2,144,40),(3,33,137),(4,138,34),(5,35,139),(6,140,36),(7,37,141),(8,142,38),(17,96,49),(18,50,89),(19,90,51),(20,52,91),(21,92,53),(22,54,93),(23,94,55),(24,56,95),(25,97,67),(26,68,98),(27,99,69),(28,70,100),(29,101,71),(30,72,102),(31,103,65),(32,66,104),(81,106,117),(82,118,107),(83,108,119),(84,120,109),(85,110,113),(86,114,111),(87,112,115),(88,116,105)], [(1,39,143),(2,144,40),(3,33,137),(4,138,34),(5,35,139),(6,140,36),(7,37,141),(8,142,38),(9,74,44),(10,45,75),(11,76,46),(12,47,77),(13,78,48),(14,41,79),(15,80,42),(16,43,73),(17,96,49),(18,50,89),(19,90,51),(20,52,91),(21,92,53),(22,54,93),(23,94,55),(24,56,95),(25,67,97),(26,98,68),(27,69,99),(28,100,70),(29,71,101),(30,102,72),(31,65,103),(32,104,66),(57,126,135),(58,136,127),(59,128,129),(60,130,121),(61,122,131),(62,132,123),(63,124,133),(64,134,125),(81,117,106),(82,107,118),(83,119,108),(84,109,120),(85,113,110),(86,111,114),(87,115,112),(88,105,116)], [(1,48,106),(2,41,107),(3,42,108),(4,43,109),(5,44,110),(6,45,111),(7,46,112),(8,47,105),(9,85,35),(10,86,36),(11,87,37),(12,88,38),(13,81,39),(14,82,40),(15,83,33),(16,84,34),(17,58,32),(18,59,25),(19,60,26),(20,61,27),(21,62,28),(22,63,29),(23,64,30),(24,57,31),(49,127,66),(50,128,67),(51,121,68),(52,122,69),(53,123,70),(54,124,71),(55,125,72),(56,126,65),(73,120,138),(74,113,139),(75,114,140),(76,115,141),(77,116,142),(78,117,143),(79,118,144),(80,119,137),(89,129,97),(90,130,98),(91,131,99),(92,132,100),(93,133,101),(94,134,102),(95,135,103),(96,136,104)], [(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 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B6C6D···6I6J···6R8A···8H12A12B12C12D12E···12L12M···12X24A···24P
order122233333344446666···66···68···81212121212···1212···1224···24
size111123366611112223···36···69···922223···36···69···9

80 irreducible representations

dim111111111111222222222266666
type++++-+-+-+-
imageC1C2C2C3C4C4C6C6C8C12C12C24S3Dic3D6Dic3C3×S3C3⋊C8C3×Dic3S3×C6C3×Dic3C3×C3⋊C8C32⋊C6C32⋊C12C2×C32⋊C6C32⋊C12He3⋊3C8
kernelC2×He3⋊3C8He3⋊3C8C2×C4×He3C2×C32⋊4C8C4×He3C22×He3C32⋊4C8C6×C12C2×He3C3×C12C62C3×C6C6×C12C3×C12C3×C12C62C2×C12C3×C6C12C12C2×C6C6C2×C4C4C4C22C2
# reps1212224284416111124222811114

Matrix representation of C2×He3⋊3C8 ►in GL8(𝔽73)

720000000
072000000
007200000
000720000
000072000
000007200
000000720
000000072
,
072000000
172000000
000720000
001720000
00001000
00000100
000000721
000000720
,
10000000
01000000
000720000
001720000
000007200
000017200
000000072
000000172
,
10000000
01000000
00000010
00000001
00100000
00010000
00001000
00000100
,
022000000
220000000
0055200000
002180000
0000552000
000021800
0000005520
000000218

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18] >;
 

C2×He3⋊3C8 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3\rtimes_3C_8
 
% in TeX
 
G:=Group("C2xHe3:3C8");
 
// GroupNames label
 
G:=SmallGroup(432,136);
 
// by ID
 
G=gap.SmallGroup(432,136);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,4037,2035,14118]);
 
// 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,e*c*e^-1=c^-1,d*e=e*d>;
 
// generators/relations
 

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