D4xC2^4 - GroupNames

G = D₄×C₂⁴ order 128 = 2⁷

Direct product of C₂⁴ and D₄

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: D₄×C₂⁴, C₄⋊C₂⁵, C₂⁶⋊₃C₂, C₂²⋊C₂⁵, C₂.₁C₂⁶, C₂³⋊₃C₂⁴, C₂⁴⋊₁₂C₂³, C₂⁵⋊₁₁C₂², (C₂×C₄)⋊₄C₂⁴, (C₂⁴×C₄)⋊₁₀C₂, (C₂²×C₄)⋊₂₅C₂³, (C₂³×C₄)⋊₆₀C₂², SmallGroup(128,2320)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂ — D₄×C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁶ — D₄×C₂⁴

Lower central

C₁ — C₂ — D₄×C₂⁴

Upper central

C₁ — C₂⁵ — D₄×C₂⁴

Jennings

C₁ — C₂ — D₄×C₂⁴

Generators and relations for D₄×C₂⁴
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 7420 in 5276 conjugacy classes, 3132 normal (5 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂×C₄, D₄, C₂³, C₂³, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂³×C₄, C₂²×D₄, C₂⁵, C₂⁵, C₂⁵, C₂⁴×C₄, D₄×C₂³, C₂⁶, D₄×C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄×C₂³, C₂⁶, D₄×C₂⁴

Smallest permutation representation of D₄×C₂⁴
►On 64 points

Generators in S₆₄

(1 17)(2 18)(3 19)(4 20)(5 24)(6 21)(7 22)(8 23)(9 28)(10 25)(11 26)(12 27)(13 32)(14 29)(15 30)(16 31)(33 50)(34 51)(35 52)(36 49)(37 54)(38 55)(39 56)(40 53)(41 58)(42 59)(43 60)(44 57)(45 64)(46 61)(47 62)(48 63)

(1 34)(2 35)(3 36)(4 33)(5 61)(6 62)(7 63)(8 64)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 51)(18 52)(19 49)(20 50)(21 47)(22 48)(23 45)(24 46)(25 59)(26 60)(27 57)(28 58)(29 55)(30 56)(31 53)(32 54)

(1 31)(2 32)(3 29)(4 30)(5 26)(6 27)(7 28)(8 25)(9 22)(10 23)(11 24)(12 21)(13 18)(14 19)(15 20)(16 17)(33 56)(34 53)(35 54)(36 55)(37 52)(38 49)(39 50)(40 51)(41 48)(42 45)(43 46)(44 47)(57 62)(58 63)(59 64)(60 61)

(1 62)(2 63)(3 64)(4 61)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)

(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)

(1 27)(2 26)(3 25)(4 28)(5 32)(6 31)(7 30)(8 29)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(33 58)(34 57)(35 60)(36 59)(37 46)(38 45)(39 48)(40 47)(41 50)(42 49)(43 52)(44 51)(53 62)(54 61)(55 64)(56 63)

G:=sub<Sym(64)| (1,17)(2,18)(3,19)(4,20)(5,24)(6,21)(7,22)(8,23)(9,28)(10,25)(11,26)(12,27)(13,32)(14,29)(15,30)(16,31)(33,50)(34,51)(35,52)(36,49)(37,54)(38,55)(39,56)(40,53)(41,58)(42,59)(43,60)(44,57)(45,64)(46,61)(47,62)(48,63), (1,34)(2,35)(3,36)(4,33)(5,61)(6,62)(7,63)(8,64)(9,41)(10,42)(11,43)(12,44)(13,37)(14,38)(15,39)(16,40)(17,51)(18,52)(19,49)(20,50)(21,47)(22,48)(23,45)(24,46)(25,59)(26,60)(27,57)(28,58)(29,55)(30,56)(31,53)(32,54), (1,31)(2,32)(3,29)(4,30)(5,26)(6,27)(7,28)(8,25)(9,22)(10,23)(11,24)(12,21)(13,18)(14,19)(15,20)(16,17)(33,56)(34,53)(35,54)(36,55)(37,52)(38,49)(39,50)(40,51)(41,48)(42,45)(43,46)(44,47)(57,62)(58,63)(59,64)(60,61), (1,62)(2,63)(3,64)(4,61)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58), (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), (1,27)(2,26)(3,25)(4,28)(5,32)(6,31)(7,30)(8,29)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(33,58)(34,57)(35,60)(36,59)(37,46)(38,45)(39,48)(40,47)(41,50)(42,49)(43,52)(44,51)(53,62)(54,61)(55,64)(56,63)>;

G:=Group( (1,17)(2,18)(3,19)(4,20)(5,24)(6,21)(7,22)(8,23)(9,28)(10,25)(11,26)(12,27)(13,32)(14,29)(15,30)(16,31)(33,50)(34,51)(35,52)(36,49)(37,54)(38,55)(39,56)(40,53)(41,58)(42,59)(43,60)(44,57)(45,64)(46,61)(47,62)(48,63), (1,34)(2,35)(3,36)(4,33)(5,61)(6,62)(7,63)(8,64)(9,41)(10,42)(11,43)(12,44)(13,37)(14,38)(15,39)(16,40)(17,51)(18,52)(19,49)(20,50)(21,47)(22,48)(23,45)(24,46)(25,59)(26,60)(27,57)(28,58)(29,55)(30,56)(31,53)(32,54), (1,31)(2,32)(3,29)(4,30)(5,26)(6,27)(7,28)(8,25)(9,22)(10,23)(11,24)(12,21)(13,18)(14,19)(15,20)(16,17)(33,56)(34,53)(35,54)(36,55)(37,52)(38,49)(39,50)(40,51)(41,48)(42,45)(43,46)(44,47)(57,62)(58,63)(59,64)(60,61), (1,62)(2,63)(3,64)(4,61)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58), (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), (1,27)(2,26)(3,25)(4,28)(5,32)(6,31)(7,30)(8,29)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(33,58)(34,57)(35,60)(36,59)(37,46)(38,45)(39,48)(40,47)(41,50)(42,49)(43,52)(44,51)(53,62)(54,61)(55,64)(56,63) );

G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,24),(6,21),(7,22),(8,23),(9,28),(10,25),(11,26),(12,27),(13,32),(14,29),(15,30),(16,31),(33,50),(34,51),(35,52),(36,49),(37,54),(38,55),(39,56),(40,53),(41,58),(42,59),(43,60),(44,57),(45,64),(46,61),(47,62),(48,63)], [(1,34),(2,35),(3,36),(4,33),(5,61),(6,62),(7,63),(8,64),(9,41),(10,42),(11,43),(12,44),(13,37),(14,38),(15,39),(16,40),(17,51),(18,52),(19,49),(20,50),(21,47),(22,48),(23,45),(24,46),(25,59),(26,60),(27,57),(28,58),(29,55),(30,56),(31,53),(32,54)], [(1,31),(2,32),(3,29),(4,30),(5,26),(6,27),(7,28),(8,25),(9,22),(10,23),(11,24),(12,21),(13,18),(14,19),(15,20),(16,17),(33,56),(34,53),(35,54),(36,55),(37,52),(38,49),(39,50),(40,51),(41,48),(42,45),(43,46),(44,47),(57,62),(58,63),(59,64),(60,61)], [(1,62),(2,63),(3,64),(4,61),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,47),(18,48),(19,45),(20,46),(21,51),(22,52),(23,49),(24,50),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58)], [(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)], [(1,27),(2,26),(3,25),(4,28),(5,32),(6,31),(7,30),(8,29),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(33,58),(34,57),(35,60),(36,59),(37,46),(38,45),(39,48),(40,47),(41,50),(42,49),(43,52),(44,51),(53,62),(54,61),(55,64),(56,63)]])

80 conjugacy classes

class	1	2A	···	2AE	2AF	···	2BK	4A	···	4P
order	1	2	···	2	2	···	2	4	···	4
size	1	1	···	1	2	···	2	2	···	2

80 irreducible representations

dim	1	1	1	1	2
type	+	+	+	+	+
image	C₁	C₂	C₂	C₂	D₄
kernel	D₄×C₂⁴	C₂⁴×C₄	D₄×C₂³	C₂⁶	C₂⁴
# reps	1	1	60	2	16

Matrix representation of D₄×C₂⁴ ►in GL₆(ℤ)

-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	1	0
0	0	0	0	0	1

-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	1	0
0	0	0	0	0	1

-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	1	0
0	0	0	0	0	1

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	1	0
0	0	0	0	0	1

-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	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	-1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1] >;

D₄×C₂⁴ in GAP, Magma, Sage, TeX

D_4\times C_2^4

% in TeX

G:=Group("D4xC2^4");

// GroupNames label

G:=SmallGroup(128,2320);

// by ID

G=gap.SmallGroup(128,2320);

# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925]);

// Polycyclic

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

// generators/relations

G = D4×C24 order 128 = 27

Direct product of C24 and D4

G = D₄×C₂⁴ order 128 = 2⁷

Direct product of C₂⁴ and D₄