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G = C2×He3⋊3Q8  order 432 = 24·33

Direct product of C2 and He3⋊3Q8

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3⋊3Q8, C62.33D6, He3⋊5(C2×Q8), (C6×C12).6C6, (C2×He3)⋊3Q8, (C3×C6)⋊3Dic6, C32⋊2(C6×Q8), C12.76(S3×C6), (C6×C12).12S3, (C3×C12).49D6, C62.9(C2×C6), C3.2(C6×Dic6), C32⋊4Q8⋊5C6, C6.10(C3×Dic6), C32⋊4(C2×Dic6), (C2×He3).19C23, (C4×He3).38C22, C32⋊C12.10C22, (C22×He3).26C22, C6.23(S3×C2×C6), (C3×C6)⋊2(C3×Q8), (C2×C4×He3).8C2, (C2×C32⋊4Q8)⋊C3, (C2×C6).53(S3×C6), (C3×C12).11(C2×C6), (C2×C12).16(C3×S3), C4.11(C2×C32⋊C6), (C2×C3⋊Dic3).3C6, C3⋊Dic3.1(C2×C6), (C3×C6).1(C22×C6), (C2×C32⋊C12).7C2, (C2×C4).4(C32⋊C6), (C3×C6).19(C22×S3), C2.3(C22×C32⋊C6), C22.8(C2×C32⋊C6), SmallGroup(432,348)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×He3⋊3Q8
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C32⋊C12 — C2×C32⋊C12 — C2×He3⋊3Q8
Lower central
C32 — C3×C6 — C2×He3⋊3Q8
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×He3⋊3Q8
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf-1=b-1, cd=dc, ce=ec, fcf-1=c-1, de=ed, df=fd, fef-1=e-1 >

Subgroups: 537 in 149 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C2×Dic6, C6×Q8, C2×He3, C2×He3, C3×Dic6, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C4×He3, C22×He3, C6×Dic6, C2×C32⋊4Q8, He3⋊3Q8, C2×C32⋊C12, C2×C4×He3, C2×He3⋊3Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×Dic6, C6×Q8, C32⋊C6, C3×Dic6, S3×C2×C6, C2×C32⋊C6, C6×Dic6, He3⋊3Q8, C22×C32⋊C6, C2×He3⋊3Q8

Smallest permutation representation of C2×He3⋊3Q8
►On 144 points
Generators in S144
(1 76)(2 73)(3 74)(4 75)(5 63)(6 64)(7 61)(8 62)(9 112)(10 109)(11 110)(12 111)(13 48)(14 45)(15 46)(16 47)(17 129)(18 130)(19 131)(20 132)(21 51)(22 52)(23 49)(24 50)(25 123)(26 124)(27 121)(28 122)(29 55)(30 56)(31 53)(32 54)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(41 86)(42 87)(43 88)(44 85)(57 93)(58 94)(59 95)(60 96)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(89 125)(90 126)(91 127)(92 128)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)

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

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

(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 114 144)(18 115 141)(19 116 142)(20 113 143)(21 34 59)(22 35 60)(23 36 57)(24 33 58)(25 128 140)(26 125 137)(27 126 138)(28 127 139)(45 66 87)(46 67 88)(47 68 85)(48 65 86)(49 72 93)(50 69 94)(51 70 95)(52 71 96)(77 107 132)(78 108 129)(79 105 130)(80 106 131)(89 101 124)(90 102 121)(91 103 122)(92 104 123)

(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)

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

62 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A6B6C6D···6I6J···6R12A12B12C12D12E···12T12U···12AB
order12223333334444446666···66···61212121212···1212···12
size111123366622181818182223···36···622226···618···18

62 irreducible representations

dim1111111122222222226666
type+++++-++-+++-
imageC1C2C2C2C3C6C6C6S3Q8D6D6C3×S3Dic6C3×Q8S3×C6S3×C6C3×Dic6C32⋊C6C2×C32⋊C6C2×C32⋊C6He3⋊3Q8
kernelC2×He3⋊3Q8He3⋊3Q8C2×C32⋊C12C2×C4×He3C2×C32⋊4Q8C32⋊4Q8C2×C3⋊Dic3C6×C12C6×C12C2×He3C3×C12C62C2×C12C3×C6C3×C6C12C2×C6C6C2×C4C4C22C2
# reps1421284212212444281214

Matrix representation of C2×He3⋊3Q8 ►in GL10(𝔽13)

1000000000
0100000000
00120000000
00012000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
12100000000
12000000000
0010000000
0001000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
00000120000
00001120000
00000001200
00000011200
00000000012
00000000112
,
3000000000
0300000000
0090000000
0009000000
0000100000
0000010000
00000012100
00000012000
00000000012
00000000112
,
3700000000
61000000000
00810000000
0005000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
5000000000
5800000000
0069000000
0067000000
0000010000
0000100000
0000000001
0000000010
0000000100
0000001000

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[5,5,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C2×He3⋊3Q8 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3\rtimes_3Q_8
 
% in TeX
 
G:=Group("C2xHe3:3Q8");
 
// GroupNames label
 
G:=SmallGroup(432,348);
 
// by ID
 
G=gap.SmallGroup(432,348);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,4037,1034,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f^-1=b^-1,c*d=d*c,c*e=e*c,f*c*f^-1=c^-1,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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