ES-(2,2):D7 - GroupNames

G = 2_-¹⁺⁴⋊D₇ order 448 = 2⁶·7

1^st semidirect product of 2_-¹⁺⁴ and D₇ acting via D₇/C₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2_-¹⁺⁴⋊₁D₇, (C₇×D₄).₃₄D₄, (C₂×C₂₈).₂₂D₄, (C₇×Q₈).₃₄D₄, D₄⋊D₁₄⋊₆C₂, C₄○D₄.₁₁D₁₄, C₂₈.₂₁₉(C₂×D₄), C₇⋊₅(D₄.₈D₄), (C₂×Q₈).₇₂D₁₄, C₁₄.₈₄C₂²≀C₂, C₂₈.₂₃D₄⋊₈C₂, D₄.₁₆(C₇⋊D₄), (C₂×C₂₈).₂₃C₂³, Q₈.₁₆(C₇⋊D₄), D₄⋊₂Dic₇⋊₁₃C₂, C₂₈.₁₀D₄⋊₁₁C₂, (C₇×2_-¹⁺⁴)⋊₁C₂, (Q₈×C₁₄).₉₉C₂², (C₂×D₂₈).₁₃₄C₂², C₂.₁₈(C₂⁴⋊D₇), (C₄×Dic₇).₆₀C₂², C₄.Dic₇.₃₀C₂², C₄.₆₆(C₂×C₇⋊D₄), (C₂×C₁₄).₄₆(C₂×D₄), (C₂×C₄).₁₃(C₇⋊D₄), (C₂×C₄).₂₃(C₂²×D₇), C₂².₁₈(C₂×C₇⋊D₄), (C₇×C₄○D₄).₂₁C₂², SmallGroup(448,779)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — 2_-¹⁺⁴⋊D₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×D₂₈ — D₄⋊D₁₄ — 2_-¹⁺⁴⋊D₇

Lower central

C₇ — C₁₄ — C₂×C₂₈ — 2_-¹⁺⁴⋊D₇

Upper central

C₁ — C₂ — C₂×C₄ — 2_-¹⁺⁴

Generators and relations for 2_-¹⁺⁴⋊D₇
G = < a,b,c,d,e,f | a⁴=b²=e⁷=f²=1, c²=d²=a², bab=faf=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd^-1=fcf=a²c, ce=ec, de=ed, fdf=a²cd, fef=e^-1 >

Subgroups: 628 in 146 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, M₄(2), D₈, SD₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄.₁₀D₄, C₄≀C₂, C₄.₄D₄, C₈⋊C₂², 2_-¹⁺⁴, C₇⋊C₈, D₂₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₇×Q₈, C₂²×D₇, D₄.₈D₄, C₄.Dic₇, C₄×Dic₇, D₁₄⋊C₄, D₄⋊D₇, Q₈⋊D₇, C₂×D₂₈, Q₈×C₁₄, Q₈×C₁₄, C₇×C₄○D₄, C₇×C₄○D₄, C₂₈.₁₀D₄, D₄⋊₂Dic₇, C₂₈.₂₃D₄, D₄⋊D₁₄, C₇×2_-¹⁺⁴, 2_-¹⁺⁴⋊D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, D₁₄, C₂²≀C₂, C₇⋊D₄, C₂²×D₇, D₄.₈D₄, C₂×C₇⋊D₄, C₂⁴⋊D₇, 2_-¹⁺⁴⋊D₇

Smallest permutation representation of 2_-¹⁺⁴⋊D₇
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	8A	8B	14A	14B	14C	14D	···	14R	28A	···	28AD
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	7	7	7	8	8	14	14	14	14	···	14	28	···	28
size	1	1	2	4	4	56	2	2	4	4	4	4	28	28	2	2	2	56	56	2	2	2	4	···	4	4	···	4

67 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

C₇⋊D₄

D₄.₈D₄

2_-¹⁺⁴⋊D₇

kernel

2_-¹⁺⁴⋊D₇

C₂₈.₁₀D₄

D₄⋊₂Dic₇

C₂₈.₂₃D₄

D₄⋊D₁₄

C₇×2_-¹⁺⁴

C₂×C₂₈

C₇×D₄

C₇×Q₈

2_-¹⁺⁴

C₂×Q₈

C₄○D₄

C₂×C₄

D₄

Q₈

C₇

C₁

# reps

Matrix representation of 2_-¹⁺⁴⋊D₇ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	0	0	0
0	0	92	98	0	0
0	0	27	0	98	0
0	0	20	73	22	15

1	0	0	0	0	0
0	1	0	0	0	0
0	0	33	44	98	0
0	0	5	108	11	88
0	0	42	52	52	2
0	0	6	1	90	33

112	0	0	0	0	0
0	112	0	0	0	0
0	0	98	0	0	0
0	0	21	15	0	0
0	0	39	88	98	0
0	0	38	56	22	15

0	1	0	0	0	0
1	0	0	0	0	0
0	0	16	73	22	30
0	0	5	27	6	92
0	0	90	41	66	27
0	0	79	80	91	4

96	47	0	0	0	0
47	96	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	1

17	66	0	0	0	0
47	96	0	0	0	0
0	0	37	69	0	0
0	0	103	76	0	0
0	0	53	56	22	30
0	0	99	53	63	91

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,92,27,20,0,0,0,98,0,73,0,0,0,0,98,22,0,0,0,0,0,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,5,42,6,0,0,44,108,52,1,0,0,98,11,52,90,0,0,0,88,2,33],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,21,39,38,0,0,0,15,88,56,0,0,0,0,98,22,0,0,0,0,0,15],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,5,90,79,0,0,73,27,41,80,0,0,22,6,66,91,0,0,30,92,27,4],[96,47,0,0,0,0,47,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,47,0,0,0,0,66,96,0,0,0,0,0,0,37,103,53,99,0,0,69,76,56,53,0,0,0,0,22,63,0,0,0,0,30,91] >;

2_-¹⁺⁴⋊D₇ in GAP, Magma, Sage, TeX

2_-^{1+4}\rtimes D_7

% in TeX

G:=Group("ES-(2,2):D7");

// GroupNames label

G:=SmallGroup(448,779);

// by ID

G=gap.SmallGroup(448,779);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,184,570,1684,851,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=e^7=f^2=1,c^2=d^2=a^2,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d^-1=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=a^2*c*d,f*e*f=e^-1>;

// generators/relations

G = 2- 1+4⋊D7 order 448 = 26·7

1st semidirect product of 2- 1+4 and D7 acting via D7/C7=C2

G = 2_-¹⁺⁴⋊D₇ order 448 = 2⁶·7

1^st semidirect product of 2_-¹⁺⁴ and D₇ acting via D₇/C₇=C₂