C2^4:3D14 - GroupNames

G = C₂⁴⋊₃D₁₄ order 448 = 2⁶·7

3^rd semidirect product of C₂⁴ and D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₃D₁₄, C₁₄.₂₆2₊¹⁺⁴, C₇⋊₂D₄², C₇⋊D₄⋊₄D₄, D₁₄⋊₆(C₂×D₄), C₂²⋊₃(D₄×D₇), (C₂×D₄)⋊₁₈D₁₄, C₂²≀C₂⋊₃D₇, Dic₇⋊₃(C₂×D₄), C₂²⋊D₂₈⋊₉C₂, C₂³⋊D₁₄⋊₄C₂, C₂₈⋊D₄⋊₁₁C₂, C₂²⋊C₄⋊₂₄D₁₄, (D₄×C₁₄)⋊₇C₂², D₁₄⋊D₄⋊₁₃C₂, D₁₄⋊C₄⋊₁₁C₂², Dic₇⋊₄D₄⋊₂C₂, Dic₇⋊D₄⋊₂C₂, (C₂×D₂₈)⋊₁₉C₂², (C₂×C₂₈).₂₈C₂³, Dic₇⋊C₄⋊₉C₂², C₁₄.₅₆(C₂²×D₄), (C₂³×D₇)⋊₇C₂², (C₂×C₁₄).₁₃₄C₂⁴, (C₂³×C₁₄)⋊₁₀C₂², (C₄×Dic₇)⋊₁₄C₂², C₂³.D₇⋊₁₅C₂², C₂.₂₈(D₄⋊₆D₁₄), (C₂²×D₇).₅₃C₂³, C₂².₁₅₅(C₂³×D₇), C₂³.₁₀₈(C₂²×D₇), (C₂²×C₁₄).₁₈₁C₂³, (C₂×Dic₇).₂₂₁C₂³, (C₂²×Dic₇)⋊₁₃C₂², (C₂×D₄×D₇)⋊₇C₂, C₂.₂₉(C₂×D₄×D₇), (C₂×C₁₄)⋊₆(C₂×D₄), (C₂×C₄×D₇)⋊₇C₂², (C₇×C₂²≀C₂)⋊₅C₂, (C₂²×C₇⋊D₄)⋊₈C₂, (C₂×C₇⋊D₄)⋊₃₉C₂², (C₇×C₂²⋊C₄)⋊₅C₂², (C₂×C₄).₂₈(C₂²×D₇), SmallGroup(448,1043)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₂⁴⋊₃D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂³×D₇ — C₂×D₄×D₇ — C₂⁴⋊₃D₁₄

Lower central

C₇ — C₂×C₁₄ — C₂⁴⋊₃D₁₄

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for C₂⁴⋊₃D₁₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁴=f²=1, ab=ba, eae^-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2476 in 428 conjugacy classes, 115 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₇, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₄×D₄, C₂²≀C₂, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂²×D₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, D₄², C₄×Dic₇, Dic₇⋊C₄, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₂²⋊C₄, C₂×C₄×D₇, C₂×D₂₈, C₂×D₂₈, D₄×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂×C₇⋊D₄, D₄×C₁₄, D₄×C₁₄, C₂³×D₇, C₂³×D₇, C₂³×C₁₄, Dic₇⋊₄D₄, C₂²⋊D₂₈, D₁₄⋊D₄, C₂³⋊D₁₄, Dic₇⋊D₄, C₂₈⋊D₄, C₇×C₂²≀C₂, C₂×D₄×D₇, C₂²×C₇⋊D₄, C₂⁴⋊₃D₁₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, 2₊¹⁺⁴, C₂²×D₇, D₄², D₄×D₇, C₂³×D₇, C₂×D₄×D₇, D₄⋊₆D₁₄, C₂⁴⋊₃D₁₄

Smallest permutation representation of C₂⁴⋊₃D₁₄
►On 112 points

Generators in S₁₁₂

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

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

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

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

(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)

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	4E	4F	4G	4H	4I	7A	7B	7C	14A	···	14I	14J	···	14AA	14AB	14AC	14AD	28A	···	28I
order	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	7	7	7	14	···	14	14	···	14	14	14	14	28	···	28
size	1	1	1	1	2	2	2	2	4	4	14	14	14	14	28	28	4	4	4	14	14	14	14	28	28	2	2	2	2	···	2	4	···	4	8	8	8	8	···	8

67 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

2₊¹⁺⁴

D₄×D₇

D₄⋊₆D₁₄

kernel

C₂⁴⋊₃D₁₄

Dic₇⋊₄D₄

C₂²⋊D₂₈

D₁₄⋊D₄

C₂³⋊D₁₄

Dic₇⋊D₄

C₂₈⋊D₄

C₇×C₂²≀C₂

C₂×D₄×D₇

C₂²×C₇⋊D₄

C₇⋊D₄

C₂²≀C₂

C₂²⋊C₄

C₂×D₄

C₂⁴

C₁₄

C₂²

C₂

# reps

Matrix representation of C₂⁴⋊₃D₁₄ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	24
0	0	0	0	0	28

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	24
0	0	0	0	0	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	28	0
0	0	0	0	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	28	0
0	0	0	0	0	28

28	0	0	0	0	0
0	1	0	0	0	0
0	0	4	10	0	0
0	0	14	28	0	0
0	0	0	0	28	0
0	0	0	0	17	1

28	0	0	0	0	0
0	28	0	0	0	0
0	0	18	7	0	0
0	0	16	11	0	0
0	0	0	0	28	0
0	0	0	0	17	1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,24,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,24,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,4,14,0,0,0,0,10,28,0,0,0,0,0,0,28,17,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,18,16,0,0,0,0,7,11,0,0,0,0,0,0,28,17,0,0,0,0,0,1] >;

C₂⁴⋊₃D₁₄ in GAP, Magma, Sage, TeX

C_2^4\rtimes_3D_{14}

% in TeX

G:=Group("C2^4:3D14");

// GroupNames label

G:=SmallGroup(448,1043);

// by ID

G=gap.SmallGroup(448,1043);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,297,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C24⋊3D14 order 448 = 26·7

3rd semidirect product of C24 and D14 acting via D14/C7=C22

G = C₂⁴⋊₃D₁₄ order 448 = 2⁶·7

3^rd semidirect product of C₂⁴ and D₁₄ acting via D₁₄/C₇=C₂²