C2^2xC4.4D4 - GroupNames

G = C₂²×C₄.₄D₄ order 128 = 2⁷

Direct product of C₂² and C₄.₄D₄

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

Aliases: C₂²×C₄.₄D₄, C₄²⋊₂₁C₂³, C₂⁵.₇₀C₂², C₂³.₁₃C₂⁴, C₂².₂₅C₂⁵, C₂⁴.₄₇₅C₂³, (Q₈×C₂³)⋊₈C₂, C₂.₉(D₄×C₂³), (C₂×Q₈)⋊₁₅C₂³, C₄.₂₅(C₂²×D₄), C₂²⋊C₄⋊₁₈C₂³, (C₂×C₄).₁₆₃C₂⁴, (C₂²×C₄²)⋊₂₁C₂, (C₂×C₄²)⋊₈₉C₂², (D₄×C₂³).₂₀C₂, (C₂²×C₄).₆₂₅D₄, C₂³.₈₉₀(C₂×D₄), (C₂×D₄).₄₄₃C₂³, (C₂²×Q₈)⋊₅₇C₂², C₂³.₃₈₀(C₄○D₄), (C₂³×C₄).₆₆₂C₂², C₂².₁₅₈(C₂²×D₄), (C₂²×C₄).₁₁₇₂C₂³, (C₂²×D₄).₅₈₂C₂², (C₂×C₄).₈₇₈(C₂×D₄), C₂.₉(C₂²×C₄○D₄), (C₂²×C₂²⋊C₄)⋊₂₉C₂, (C₂×C₂²⋊C₄)⋊₈₄C₂², C₂².₁₄₉(C₂×C₄○D₄), SmallGroup(128,2168)

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

Derived series

C₁ — C₂² — C₂²×C₄.₄D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄² — C₂²×C₄.₄D₄

Lower central

C₁ — C₂² — C₂²×C₄.₄D₄

Upper central

C₁ — C₂⁴ — C₂²×C₄.₄D₄

Jennings

C₁ — C₂² — C₂²×C₄.₄D₄

Generators and relations for C₂²×C₄.₄D₄
G = < a,b,c,d,e | a²=b²=c⁴=d⁴=1, e²=c², ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=c²d^-1 >

Subgroups: 1580 in 940 conjugacy classes, 476 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₄.₄D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂⁵, C₂²×C₄², C₂²×C₂²⋊C₄, C₂×C₄.₄D₄, D₄×C₂³, Q₈×C₂³, C₂²×C₄.₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, C₂⁵, C₂×C₄.₄D₄, D₄×C₂³, C₂²×C₄○D₄, C₂²×C₄.₄D₄

Smallest permutation representation of C₂²×C₄.₄D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2W	4A	···	4X	4Y	···	4AF
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	2	2
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄
kernel	C₂²×C₄.₄D₄	C₂²×C₄²	C₂²×C₂²⋊C₄	C₂×C₄.₄D₄	D₄×C₂³	Q₈×C₂³	C₂²×C₄	C₂³
# reps	1	1	4	24	1	1	8	16

Matrix representation of C₂²×C₄.₄D₄ ►in GL₇(𝔽₅)

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

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

4	0	0	0	0	0	0
0	2	0	0	0	0	0
0	4	3	0	0	0	0
0	0	0	3	4	0	0
0	0	0	0	2	0	0
0	0	0	0	0	2	0
0	0	0	0	0	1	3

4	0	0	0	0	0	0
0	2	0	0	0	0	0
0	0	2	0	0	0	0
0	0	0	3	0	0	0
0	0	0	0	3	0	0
0	0	0	0	0	1	0
0	0	0	0	0	3	4

1	0	0	0	0	0	0
0	3	2	0	0	0	0
0	0	2	0	0	0	0
0	0	0	3	0	0	0
0	0	0	3	2	0	0
0	0	0	0	0	1	1
0	0	0	0	0	3	4

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,3,0,0,0,0,0,1,4] >;

C₂²×C₄.₄D₄ in GAP, Magma, Sage, TeX

C_2^2\times C_4._4D_4

% in TeX

G:=Group("C2^2xC4.4D4");

// GroupNames label

G:=SmallGroup(128,2168);

// by ID

G=gap.SmallGroup(128,2168);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184]);

// Polycyclic

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

// generators/relations

G = C22×C4.4D4 order 128 = 27

Direct product of C22 and C4.4D4

G = C₂²×C₄.₄D₄ order 128 = 2⁷

Direct product of C₂² and C₄.₄D₄