D14:M4(2) - GroupNames

G = D₁₄⋊M₄(2) order 448 = 2⁶·7

1^st semidirect product of D₁₄ and M₄(2) acting via M₄(2)/C₂×C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₄⋊₅M₄(2), (C₂×C₈)⋊₁₈D₁₄, D₁₄⋊C₈⋊₁₇C₂, C₂²⋊C₈⋊₁₁D₇, (C₄×D₇).₄₆D₄, C₄.₁₉₅(D₄×D₇), (C₂×C₅₆)⋊₂₀C₂², C₂₈.₃₅₄(C₂×D₄), (C₂×C₁₄)⋊₁M₄(2), (C₂³×D₇).₅C₄, C₂³.₄₇(C₄×D₇), C₂²⋊₃(C₈⋊D₇), C₇⋊₁(C₂⁴.₄C₄), C₂.₁₂(D₇×M₄(2)), C₁₄.₅(C₂×M₄(2)), C₂₈.₅₅D₄⋊₂₃C₂, (C₂×C₂₈).₈₂₁C₂³, (C₂²×C₄).₃₀₄D₁₄, D₁₄.₁₀(C₂²⋊C₄), (C₂²×Dic₇).₁₀C₄, Dic₇.₁₁(C₂²⋊C₄), (C₂²×C₂₈).₃₃₈C₂², (C₂×C₄×D₇).₁₈C₄, (C₂×C₇⋊C₈)⋊₂₇C₂², C₂.₉(C₂×C₈⋊D₇), C₂.₉(D₇×C₂²⋊C₄), (C₂×C₈⋊D₇)⋊₁₁C₂, (C₇×C₂²⋊C₈)⋊₁₅C₂, (C₂×C₄).₁₃₂(C₄×D₇), C₁₄.₈(C₂×C₂²⋊C₄), (D₇×C₂²×C₄).₁₇C₂, C₂².₁₀₃(C₂×C₄×D₇), (C₂×C₂₈).₁₅₄(C₂×C₄), (C₂×C₄×D₇).₂₇₃C₂², (C₂²×C₁₄).₃₉(C₂×C₄), (C₂×C₁₄).₇₆(C₂²×C₄), (C₂×Dic₇).₈₅(C₂×C₄), (C₂²×D₇).₅₅(C₂×C₄), (C₂×C₄).₇₆₃(C₂²×D₇), SmallGroup(448,260)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₁₄⋊M₄(2)

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×C₄×D₇ — D₇×C₂²×C₄ — D₁₄⋊M₄(2)

Lower central

C₇ — C₂×C₁₄ — D₁₄⋊M₄(2)

Upper central

C₁ — C₂×C₄ — C₂²⋊C₈

Generators and relations for D₁₄⋊M₄(2)
G = < a,b,c,d | a¹⁴=b²=c⁸=d²=1, bab=cac^-1=dad=a^-1, cbc^-1=a⁵b, dbd=a¹²b, dcd=c⁵ >

Subgroups: 956 in 190 conjugacy classes, 61 normal (33 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, C₂²⋊C₈, C₂×M₄(2), C₂³×C₄, C₇⋊C₈, C₅₆, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂⁴.₄C₄, C₈⋊D₇, C₂×C₇⋊C₈, C₂×C₅₆, C₂×C₄×D₇, C₂×C₄×D₇, C₂²×Dic₇, C₂²×C₂₈, C₂³×D₇, D₁₄⋊C₈, C₂₈.₅₅D₄, C₇×C₂²⋊C₈, C₂×C₈⋊D₇, D₇×C₂²×C₄, D₁₄⋊M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₇, C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, D₁₄, C₂×C₂²⋊C₄, C₂×M₄(2), C₄×D₇, C₂²×D₇, C₂⁴.₄C₄, C₈⋊D₇, C₂×C₄×D₇, D₄×D₇, D₇×C₂²⋊C₄, C₂×C₈⋊D₇, D₇×M₄(2), D₁₄⋊M₄(2)

Smallest permutation representation of D₁₄⋊M₄(2)
►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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 42)(11 41)(12 40)(13 39)(14 38)(15 85)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(43 74)(44 73)(45 72)(46 71)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 99)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)

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

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

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

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

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

88 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	7A	7B	7C	8A	8B	8C	8D	8E	8F	8G	8H	14A	···	14I	14J	···	14O	28A	···	28L	28M	···	28R	56A	···	56X
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	7	7	7	8	8	8	8	8	8	8	8	14	···	14	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	2	2	14	14	14	14	1	1	1	1	2	2	14	14	14	14	2	2	2	4	4	4	4	28	28	28	28	2	···	2	4	···	4	2	···	2	4	···	4	4	···	4

88 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+				+	+			+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₇	M₄(2)	M₄(2)	D₁₄	D₁₄	C₄×D₇	C₄×D₇	C₈⋊D₇	D₄×D₇	D₇×M₄(2)
kernel	D₁₄⋊M₄(2)	D₁₄⋊C₈	C₂₈.₅₅D₄	C₇×C₂²⋊C₈	C₂×C₈⋊D₇	D₇×C₂²×C₄	C₂×C₄×D₇	C₂²×Dic₇	C₂³×D₇	C₄×D₇	C₂²⋊C₈	D₁₄	C₂×C₁₄	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₄	C₂
# reps	1	2	1	1	2	1	4	2	2	4	3	4	4	6	3	6	6	24	6	6

Matrix representation of D₁₄⋊M₄(2) ►in GL₆(𝔽₁₁₃)

89	103	0	0	0	0
10	103	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	112	0	0	0
0	0	26	1	0	0
0	0	0	0	112	0
0	0	0	0	0	112

10	24	0	0	0	0
10	103	0	0	0	0
0	0	100	112	0	0
0	0	41	13	0	0
0	0	0	0	0	112
0	0	0	0	15	0

103	89	0	0	0	0
103	10	0	0	0	0
0	0	1	0	0	0
0	0	87	112	0	0
0	0	0	0	112	0
0	0	0	0	0	1

G:=sub<GL(6,GF(113))| [89,10,0,0,0,0,103,103,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,112,26,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[10,10,0,0,0,0,24,103,0,0,0,0,0,0,100,41,0,0,0,0,112,13,0,0,0,0,0,0,0,15,0,0,0,0,112,0],[103,103,0,0,0,0,89,10,0,0,0,0,0,0,1,87,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1] >;

D₁₄⋊M₄(2) in GAP, Magma, Sage, TeX

D_{14}\rtimes M_4(2)

% in TeX

G:=Group("D14:M4(2)");

// GroupNames label

G:=SmallGroup(448,260);

// by ID

G=gap.SmallGroup(448,260);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,58,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^5*b,d*b*d=a^12*b,d*c*d=c^5>;

// generators/relations

G = D14⋊M4(2) order 448 = 26·7

1st semidirect product of D14 and M4(2) acting via M4(2)/C2×C4=C2

G = D₁₄⋊M₄(2) order 448 = 2⁶·7

1^st semidirect product of D₁₄ and M₄(2) acting via M₄(2)/C₂×C₄=C₂