C2^3:2Dic14 - GroupNames

G = C₂³⋊₂Dic₁₄ order 448 = 2⁶·7

2^nd semidirect product of C₂³ and Dic₁₄ acting via Dic₁₄/C₁₄=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³⋊₂Dic₁₄, C₂⁴.₂₃D₁₄, C₁₄.₁2₊¹⁺⁴, (C₂²×C₁₄)⋊₅Q₈, C₄⋊Dic₇⋊₃C₂², C₇⋊₁(C₂³⋊₂Q₈), C₂₈.₄₈D₄⋊₃C₂, C₁₄.₆(C₂²×Q₈), (C₂×C₁₄).₂₇C₂⁴, Dic₇⋊C₄⋊₁C₂², C₂²⋊C₄.₈₆D₁₄, C₂.₆(D₄⋊₆D₁₄), (C₂×C₂₈).₁₂₇C₂³, C₂²⋊Dic₁₄⋊₁C₂, (C₂×Dic₁₄)⋊₂C₂², (C₂²×C₄).₁₆₉D₁₄, (C₂×Dic₇).₈C₂³, C₂².₅(C₂×Dic₁₄), C₂.₈(C₂²×Dic₁₄), C₂².₆₉(C₂³×D₇), (C₂³×C₁₄).₅₃C₂², (C₂²×C₂₈).₇₁C₂², C₂³.₁₄₄(C₂²×D₇), C₂³.D₇.₈₅C₂², (C₂²×C₁₄).₁₁₉C₂³, (C₂²×Dic₇).₇₆C₂², (C₂×C₁₄).₄₉(C₂×Q₈), (C₂×C₂²⋊C₄).₁₈D₇, (C₁₄×C₂²⋊C₄).₁₈C₂, (C₂×C₄).₁₃₃(C₂²×D₇), (C₂×C₂³.D₇).₂₂C₂, (C₇×C₂²⋊C₄).₉₇C₂², SmallGroup(448,936)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₂³⋊₂Dic₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₂²×Dic₇ — C₂×C₂³.D₇ — C₂³⋊₂Dic₁₄

Lower central

C₇ — C₂×C₁₄ — C₂³⋊₂Dic₁₄

Upper central

C₁ — C₂² — C₂×C₂²⋊C₄

Generators and relations for C₂³⋊₂Dic₁₄
G = < a,b,c,d,e | a²=b²=c²=d²⁸=1, e²=d¹⁴, ab=ba, dad^-1=ac=ca, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 980 in 242 conjugacy classes, 111 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₁₄, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²⋊Q₈, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂³⋊₂Q₈, Dic₇⋊C₄, C₄⋊Dic₇, C₂³.D₇, C₇×C₂²⋊C₄, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂³×C₁₄, C₂²⋊Dic₁₄, C₂₈.₄₈D₄, C₂×C₂³.D₇, C₁₄×C₂²⋊C₄, C₂³⋊₂Dic₁₄
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₂⁴, D₁₄, C₂²×Q₈, 2₊¹⁺⁴, Dic₁₄, C₂²×D₇, C₂³⋊₂Q₈, C₂×Dic₁₄, C₂³×D₇, C₂²×Dic₁₄, D₄⋊₆D₁₄, C₂³⋊₂Dic₁₄

Smallest permutation representation of C₂³⋊₂Dic₁₄
►On 112 points

Generators in S₁₁₂

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

(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)

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

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

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

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

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

82 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4L	7A	7B	7C	14A	···	14U	14V	···	14AG	28A	···	28X
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	1	1	2	···	2	4	4	4	4	28	···	28	2	2	2	2	···	2	4	···	4	4	···	4

82 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	-	+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	Q₈	D₇	D₁₄	D₁₄	D₁₄	Dic₁₄	2₊¹⁺⁴	D₄⋊₆D₁₄
kernel	C₂³⋊₂Dic₁₄	C₂²⋊Dic₁₄	C₂₈.₄₈D₄	C₂×C₂³.D₇	C₁₄×C₂²⋊C₄	C₂²×C₁₄	C₂×C₂²⋊C₄	C₂²⋊C₄	C₂²×C₄	C₂⁴	C₂³	C₁₄	C₂
# reps	1	8	4	2	1	4	3	12	6	3	24	2	12

Matrix representation of C₂³⋊₂Dic₁₄ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	0	0	0
0	0	15	1	0	0
0	0	0	0	28	0
0	0	22	0	0	1

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	22	0	28	0
0	0	7	0	0	28

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

0	1	0	0	0	0
28	0	0	0	0	0
0	0	2	8	0	0
0	0	22	27	0	0
0	0	20	1	0	13
0	0	5	28	16	0

0	17	0	0	0	0
17	0	0	0	0	0
0	0	13	0	12	0
0	0	15	0	26	1
0	0	10	0	16	0
0	0	9	28	13	0

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,15,0,22,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,22,7,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,2,22,20,5,0,0,8,27,1,28,0,0,0,0,0,16,0,0,0,0,13,0],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,13,15,10,9,0,0,0,0,0,28,0,0,12,26,16,13,0,0,0,1,0,0] >;

C₂³⋊₂Dic₁₄ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_{14}

% in TeX

G:=Group("C2^3:2Dic14");

// GroupNames label

G:=SmallGroup(448,936);

// by ID

G=gap.SmallGroup(448,936);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,758,675,570,80,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=d^14,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23⋊2Dic14 order 448 = 26·7

2nd semidirect product of C23 and Dic14 acting via Dic14/C14=C22

G = C₂³⋊₂Dic₁₄ order 448 = 2⁶·7

2^nd semidirect product of C₂³ and Dic₁₄ acting via Dic₁₄/C₁₄=C₂²