C2^4:Dic7 - GroupNames

G = C₂⁴⋊Dic₇ order 448 = 2⁶·7

1^st semidirect product of C₂⁴ and Dic₇ acting via Dic₇/C₇=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₁Dic₇, C₇⋊₃C₂≀C₄, (C₂×C₂₈).₃D₄, (C₂³×C₁₄)⋊₃C₄, (C₂×D₄).₇D₁₄, C₂₈.D₄⋊₂C₂, C₂²≀C₂.₁D₇, C₂²⋊C₄⋊₁Dic₇, C₂³⋊Dic₇⋊₂C₂, (D₄×C₁₄).₅C₂², (C₂²×C₁₄).₁₄D₄, C₂³.₅(C₇⋊D₄), C₂³.₁(C₂×Dic₇), C₁₄.₁₉(C₂³⋊C₄), C₂.₄(C₂³⋊Dic₇), C₂².₁₂(C₂³.D₇), (C₇×C₂²⋊C₄)⋊₁C₄, (C₂×C₄).₅(C₇⋊D₄), (C₇×C₂²≀C₂).₁C₂, (C₂²×C₁₄).₁₃(C₂×C₄), (C₂×C₁₄).₉₄(C₂²⋊C₄), SmallGroup(448,93)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂⁴⋊Dic₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — D₄×C₁₄ — C₂³⋊Dic₇ — C₂⁴⋊Dic₇

Lower central

C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂⁴⋊Dic₇

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂²≀C₂

Generators and relations for C₂⁴⋊Dic₇
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁴=1, f²=e⁷, ab=ba, eae^-1=ac=ca, ad=da, faf^-1=abcd, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 396 in 94 conjugacy classes, 23 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₇⋊C₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂²×C₁₄, C₂≀C₄, C₄.Dic₇, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₂²⋊C₄, D₄×C₁₄, D₄×C₁₄, C₂³×C₁₄, C₂₈.D₄, C₂³⋊Dic₇, C₇×C₂²≀C₂, C₂⁴⋊Dic₇
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₇, C₂²⋊C₄, Dic₇, D₁₄, C₂³⋊C₄, C₂×Dic₇, C₇⋊D₄, C₂≀C₄, C₂³.D₇, C₂³⋊Dic₇, C₂⁴⋊Dic₇

Smallest permutation representation of C₂⁴⋊Dic₇
►On 56 points

Generators in S₅₆

(8 39)(9 33)(10 41)(11 35)(12 29)(13 37)(14 31)(22 36)(23 30)(24 38)(25 32)(26 40)(27 34)(28 42)

(1 45)(2 53)(3 47)(4 55)(5 49)(6 43)(7 51)(8 39)(9 33)(10 41)(11 35)(12 29)(13 37)(14 31)(15 48)(16 56)(17 50)(18 44)(19 52)(20 46)(21 54)(22 36)(23 30)(24 38)(25 32)(26 40)(27 34)(28 42)

(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)

(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)

(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)

(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 15)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 44 36 51)(30 43 37 50)(31 56 38 49)(32 55 39 48)(33 54 40 47)(34 53 41 46)(35 52 42 45)

G:=sub<Sym(56)| (8,39)(9,33)(10,41)(11,35)(12,29)(13,37)(14,31)(22,36)(23,30)(24,38)(25,32)(26,40)(27,34)(28,42), (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(8,39)(9,33)(10,41)(11,35)(12,29)(13,37)(14,31)(15,48)(16,56)(17,50)(18,44)(19,52)(20,46)(21,54)(22,36)(23,30)(24,38)(25,32)(26,40)(27,34)(28,42), (8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,44,36,51)(30,43,37,50)(31,56,38,49)(32,55,39,48)(33,54,40,47)(34,53,41,46)(35,52,42,45)>;

G:=Group( (8,39)(9,33)(10,41)(11,35)(12,29)(13,37)(14,31)(22,36)(23,30)(24,38)(25,32)(26,40)(27,34)(28,42), (1,45)(2,53)(3,47)(4,55)(5,49)(6,43)(7,51)(8,39)(9,33)(10,41)(11,35)(12,29)(13,37)(14,31)(15,48)(16,56)(17,50)(18,44)(19,52)(20,46)(21,54)(22,36)(23,30)(24,38)(25,32)(26,40)(27,34)(28,42), (8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,15)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,44,36,51)(30,43,37,50)(31,56,38,49)(32,55,39,48)(33,54,40,47)(34,53,41,46)(35,52,42,45) );

G=PermutationGroup([[(8,39),(9,33),(10,41),(11,35),(12,29),(13,37),(14,31),(22,36),(23,30),(24,38),(25,32),(26,40),(27,34),(28,42)], [(1,45),(2,53),(3,47),(4,55),(5,49),(6,43),(7,51),(8,39),(9,33),(10,41),(11,35),(12,29),(13,37),(14,31),(15,48),(16,56),(17,50),(18,44),(19,52),(20,46),(21,54),(22,36),(23,30),(24,38),(25,32),(26,40),(27,34),(28,42)], [(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,15),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,44,36,51),(30,43,37,50),(31,56,38,49),(32,55,39,48),(33,54,40,47),(34,53,41,46),(35,52,42,45)]])

55 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	7A	7B	7C	8A	8B	14A	···	14I	14J	···	14AA	14AB	14AC	14AD	28A	···	28I
order	1	2	2	2	2	2	2	4	4	4	4	7	7	7	8	8	14	···	14	14	···	14	14	14	14	28	···	28
size	1	1	2	4	4	4	4	4	8	56	56	2	2	2	56	56	2	···	2	4	···	4	8	8	8	8	···	8

55 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+	-	+	-			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₇	Dic₇	D₁₄	Dic₇	C₇⋊D₄	C₇⋊D₄	C₂³⋊C₄	C₂≀C₄	C₂³⋊Dic₇	C₂⁴⋊Dic₇
kernel	C₂⁴⋊Dic₇	C₂₈.D₄	C₂³⋊Dic₇	C₇×C₂²≀C₂	C₇×C₂²⋊C₄	C₂³×C₁₄	C₂×C₂₈	C₂²×C₁₄	C₂²≀C₂	C₂²⋊C₄	C₂×D₄	C₂⁴	C₂×C₄	C₂³	C₁₄	C₇	C₂	C₁
# reps	1	1	1	1	2	2	1	1	3	3	3	3	6	6	1	2	6	12

Matrix representation of C₂⁴⋊Dic₇ ►in GL₆(𝔽₁₁₃)

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	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	91	0	0
0	0	0	112	0	0
0	0	0	55	0	1
0	0	0	55	1	0

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	108	0	112	0
0	0	108	0	0	112

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

7	0	0	0	0	0
50	97	0	0	0	0
0	0	1	0	0	0
0	0	72	112	0	0
0	0	0	0	1	0
0	0	108	0	0	112

108	104	0	0	0	0
28	5	0	0	0	0
0	0	58	0	91	0
0	0	0	0	112	1
0	0	45	0	55	0
0	0	4	112	55	0

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,91,112,55,55,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,108,108,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[7,50,0,0,0,0,0,97,0,0,0,0,0,0,1,72,0,108,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[108,28,0,0,0,0,104,5,0,0,0,0,0,0,58,0,45,4,0,0,0,0,0,112,0,0,91,112,55,55,0,0,0,1,0,0] >;

C₂⁴⋊Dic₇ in GAP, Magma, Sage, TeX

C_2^4\rtimes {\rm Dic}_7

% in TeX

G:=Group("C2^4:Dic7");

// GroupNames label

G:=SmallGroup(448,93);

// by ID

G=gap.SmallGroup(448,93);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,675,297,1684,18822]);

// Polycyclic

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

// generators/relations

G = C24⋊Dic7 order 448 = 26·7

1st semidirect product of C24 and Dic7 acting via Dic7/C7=C4

G = C₂⁴⋊Dic₇ order 448 = 2⁶·7

1^st semidirect product of C₂⁴ and Dic₇ acting via Dic₇/C₇=C₄