C2^3xC4oD4 - GroupNames

G = C₂³×C₄○D₄ order 128 = 2⁷

Direct product of C₂³ and C₄○D₄

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

Aliases: C₂³×C₄○D₄, D₄⋊₃C₂⁴, C₂.₃C₂⁶, Q₈⋊₃C₂⁴, C₄.₂₃C₂⁵, C₂².₁C₂⁵, C₂⁵.₉₇C₂², C₂³.₂₈₂C₂⁴, C₂⁴.₆₆₁C₂³, D₄ ○(C₂³×C₄), C₄ ○(D₄×C₂³), C₄ ○(Q₈×C₂³), Q₈ ○(C₂³×C₄), (C₂×C₄)⋊₅C₂⁴, (C₂⁴×C₄)⋊₁₁C₂, (D₄×C₂³)⋊₂₀C₂, (C₂×D₄)⋊₂₅C₂³, (C₂×Q₈)⋊₂₆C₂³, (Q₈×C₂³)⋊₁₆C₂, (C₂³×C₄)⋊₆₁C₂², (C₂²×C₄)⋊₂₆C₂³, (C₂²×D₄)⋊₆₈C₂², (C₂²×Q₈)⋊₇₃C₂², C₄ ○(C₂²×C₄○D₄), (C₂×Q₈)○(C₂³×C₄), (C₂×C₄)○(Q₈×C₂³), (C₂×C₄)○₂(C₂²×D₄), (C₂²×C₄)○₂(C₂×D₄), (C₂²×C₄)○₂(C₂×Q₈), (C₂×C₄)○₂(C₂²×Q₈), (C₂²×C₄)○(C₄○D₄), (C₂³×C₄)○(Q₈×C₂³), (C₂²×C₄)○(Q₈×C₂³), (C₂³×C₄)○(C₂²×Q₈), (C₂²×C₄)○₂(C₂²×Q₈), (C₂×C₄)○₂(C₂×C₄○D₄), (C₂×C₄)○(C₂²×C₄○D₄), (C₂²×C₄)○(C₂×C₄○D₄), (C₂²×C₄)○(C₂²×C₄○D₄), SmallGroup(128,2322)

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

Derived series

C₁ — C₂ — C₂³×C₄○D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂⁴×C₄ — C₂³×C₄○D₄

Lower central

C₁ — C₂ — C₂³×C₄○D₄

Upper central

C₁ — C₂³×C₄ — C₂³×C₄○D₄

Jennings

C₁ — C₂ — C₂³×C₄○D₄

Generators and relations for C₂³×C₄○D₄
G = < a,b,c,d,e,f | a²=b²=c²=d⁴=f²=1, e²=d², 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=d²e >

Subgroups: 3644 in 3260 conjugacy classes, 2876 normal (6 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂⁵, C₂⁴×C₄, D₄×C₂³, Q₈×C₂³, C₂²×C₄○D₄, C₂³×C₄○D₄
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, C₂⁵, C₂²×C₄○D₄, C₂⁶, C₂³×C₄○D₄

Smallest permutation representation of C₂³×C₄○D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	···	2O	2P	···	2AM	4A	···	4P	4Q	···	4AN
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	1	···	1	2	···	2

80 irreducible representations

dim	1	1	1	1	1	2
type	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄○D₄
kernel	C₂³×C₄○D₄	C₂⁴×C₄	D₄×C₂³	Q₈×C₂³	C₂²×C₄○D₄	C₂³
# reps	1	3	3	1	56	16

Matrix representation of C₂³×C₄○D₄ ►in GL₅(𝔽₅)

4	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	4	0
0	0	0	0	4

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

4	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	2	0
0	0	0	0	2

1	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	0	1
0	0	0	4	0

4	0	0	0	0
0	4	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,1,0],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₂³×C₄○D₄ in GAP, Magma, Sage, TeX

C_2^3\times C_4\circ D_4

% in TeX

G:=Group("C2^3xC4oD4");

// GroupNames label

G:=SmallGroup(128,2322);

// by ID

G=gap.SmallGroup(128,2322);

# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925,352]);

// Polycyclic

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

// generators/relations

G = C23×C4○D4 order 128 = 27

Direct product of C23 and C4○D4

G = C₂³×C₄○D₄ order 128 = 2⁷

Direct product of C₂³ and C₄○D₄