C14.38ES+(2,2) - GroupNames

G = C₁₄.₃₈2₊¹⁺⁴ order 448 = 2⁶·7

38^th non-split extension by C₁₄ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₄.₃₈2₊¹⁺⁴, (C₄×D₇)⋊₁D₄, C₄⋊C₄⋊₂₂D₁₄, (C₂×D₄)⋊₇D₁₄, C₄⋊D₄⋊₁₀D₇, C₄.₁₈₃(D₄×D₇), C₂³⋊D₁₄⋊₉C₂, C₂₈⋊D₄⋊₁₆C₂, C₂₈⋊₇D₄⋊₃₃C₂, C₂₈⋊₂D₄⋊₁₈C₂, C₄⋊D₂₈⋊₂₁C₂, C₂²⋊C₄⋊₁₀D₁₄, D₁₄.₄₁(C₂×D₄), C₂₈.₂₂₇(C₂×D₄), (C₂²×C₄)⋊₁₆D₁₄, Dic₇.₆(C₂×D₄), C₂²⋊D₂₈⋊₁₂C₂, D₁₄⋊C₄⋊₁₇C₂², (D₄×C₁₄)⋊₁₂C₂², (C₂×D₂₈)⋊₂₃C₂², C₄⋊Dic₇⋊₃₁C₂², C₁₄.₆₆(C₂²×D₄), (C₂×C₂₈).₁₇₃C₂³, (C₂×C₁₄).₁₅₁C₂⁴, (C₂²×C₂₈)⋊₂₀C₂², C₇⋊₃(C₂².₂₉C₂⁴), (C₄×Dic₇)⋊₂₁C₂², C₂.₄₀(D₄⋊₆D₁₄), C₂.₂₇(D₄⋊₈D₁₄), C₂³.D₇⋊₂₃C₂², Dic₇.D₄⋊₁₈C₂, (C₂×Dic₁₄)⋊₅₄C₂², (C₂²×C₁₄).₂₀C₂³, (C₂×Dic₇).₇₂C₂³, (C₂³×D₇).₄₆C₂², C₂³.₁₁₂(C₂²×D₇), C₂².₁₇₂(C₂³×D₇), (C₂²×D₇).₁₈₆C₂³, (C₂×D₄×D₇)⋊₁₀C₂, C₂.₃₉(C₂×D₄×D₇), (C₂×C₄×D₇)⋊₁₃C₂², C₄⋊C₄⋊₇D₇⋊₂₀C₂, (C₂×C₄○D₂₈)⋊₂₀C₂, (C₇×C₄⋊D₄)⋊₁₃C₂, (C₇×C₄⋊C₄)⋊₁₀C₂², (C₂×C₇⋊D₄)⋊₁₄C₂², (C₂×C₄).₃₈(C₂²×D₇), (C₇×C₂²⋊C₄)⋊₁₂C₂², SmallGroup(448,1060)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₁₄.₃₈2₊¹⁺⁴

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂³×D₇ — C₂×D₄×D₇ — C₁₄.₃₈2₊¹⁺⁴

Lower central

C₇ — C₂×C₁₄ — C₁₄.₃₈2₊¹⁺⁴

Upper central

C₁ — C₂² — C₄⋊D₄

Generators and relations for C₁₄.₃₈2₊¹⁺⁴
G = < a,b,c,d,e | a¹⁴=b⁴=c²=e²=1, d²=b², ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=b^-1, bd=db, be=eb, dcd^-1=ece=a⁷c, ede=b²d >

Subgroups: 1996 in 334 conjugacy classes, 103 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄²⋊C₂, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂²×D₄, C₂×C₄○D₄, Dic₁₄, C₄×D₇, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂².₂₉C₂⁴, C₄×Dic₇, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, C₂×D₂₈, C₄○D₂₈, D₄×D₇, C₂×C₇⋊D₄, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, D₄×C₁₄, C₂³×D₇, C₂²⋊D₂₈, Dic₇.D₄, C₄⋊C₄⋊₇D₇, C₄⋊D₂₈, C₂₈⋊₇D₄, C₂³⋊D₁₄, C₂₈⋊₂D₄, C₂₈⋊D₄, C₇×C₄⋊D₄, C₂×C₄○D₂₈, C₂×D₄×D₇, C₁₄.₃₈2₊¹⁺⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, 2₊¹⁺⁴, C₂²×D₇, C₂².₂₉C₂⁴, D₄×D₇, C₂³×D₇, C₂×D₄×D₇, D₄⋊₆D₁₄, D₄⋊₈D₁₄, C₁₄.₃₈2₊¹⁺⁴

Smallest permutation representation of C₁₄.₃₈2₊¹⁺⁴
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	7A	7B	7C	14A	···	14I	14J	···	14O	14P	···	14U	28A	···	28L	28M	···	28R
order	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	7	7	7	14	···	14	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	4	4	4	14	14	28	28	28	2	2	4	4	4	14	14	28	28	28	2	2	2	2	···	2	4	···	4	8	···	8	4	···	4	8	···	8

64 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

2₊¹⁺⁴

D₄×D₇

D₄⋊₆D₁₄

D₄⋊₈D₁₄

kernel

C₁₄.₃₈2₊¹⁺⁴

C₂²⋊D₂₈

Dic₇.D₄

C₄⋊C₄⋊₇D₇

C₄⋊D₂₈

C₂₈⋊₇D₄

C₂³⋊D₁₄

C₂₈⋊₂D₄

C₂₈⋊D₄

C₇×C₄⋊D₄

C₂×C₄○D₂₈

C₂×D₄×D₇

C₄×D₇

C₄⋊D₄

C₂²⋊C₄

C₄⋊C₄

C₂²×C₄

C₂×D₄

C₁₄

C₄

C₂

# reps

Matrix representation of C₁₄.₃₈2₊¹⁺⁴ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	25	25	0	0
0	0	4	11	0	0
0	0	0	0	25	25
0	0	0	0	4	11

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

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

1	27	0	0	0	0
0	28	0	0	0	0
0	0	18	27	0	0
0	0	3	11	0	0
0	0	0	0	18	27
0	0	0	0	3	11

1	27	0	0	0	0
0	28	0	0	0	0
0	0	18	27	0	0
0	0	2	11	0	0
0	0	0	0	18	27
0	0	0	0	2	11

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,25,4,0,0,0,0,25,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,18,3,0,0,0,0,27,11,0,0,0,0,0,0,18,3,0,0,0,0,27,11],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,0,0,0,0,18,2,0,0,0,0,27,11] >;

C₁₄.₃₈2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_{14}._{38}2_+^{1+4}

% in TeX

G:=Group("C14.38ES+(2,2)");

// GroupNames label

G:=SmallGroup(448,1060);

// by ID

G=gap.SmallGroup(448,1060);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,570,297,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=b^2*d>;

// generators/relations

G = C14.382+ 1+4 order 448 = 26·7

38th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

G = C₁₄.₃₈2₊¹⁺⁴ order 448 = 2⁶·7

38^th non-split extension by C₁₄ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂