C2^2xES-(2,2)

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

Aliases: C₂²×2_-⁽¹⁺⁴⁾, C₂.₅C₂⁶, C₄.₁₆C₂⁵, D₄.₁₁C₂⁴, Q₈.₁₁C₂⁴, C₂².₁₉C₂⁵, C₂³.₂₈₃C₂⁴, C₂⁴.₆₂₁C₂³, Q₈ ○(C₂²×D₄), D₄ ○(C₂²×Q₈), C₄○D₄⋊₈C₂³, (C₂×Q₈)⋊₂₃C₂³, (Q₈×C₂³)⋊₁₅C₂, (C₂×C₄).₆₁₈C₂⁴, (C₂×D₄).₅₀₉C₂³, (C₂²×Q₈)⋊₇₁C₂², (C₂³×C₄).₆₂₅C₂², (C₂²×C₄).₁₅₉₄C₂³, (C₂²×D₄).₆₁₇C₂², (C₂×D₄)○₂(C₂×Q₈), (C₂×Q₈)○₂(C₂²×D₄), (C₂×C₄○D₄)⋊₈₀C₂², (C₂²×C₄○D₄)⋊₂₉C₂, SmallGroup(128,2324)

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

Derived series

C₁ — C₂ — C₂²×2_-⁽¹⁺⁴⁾

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — Q₈×C₂³ — C₂²×2_-⁽¹⁺⁴⁾

Lower central

C₁ — C₂ — C₂²×2_-⁽¹⁺⁴⁾

Upper central

C₁ — C₂³ — C₂²×2_-⁽¹⁺⁴⁾

Jennings

C₁ — C₂ — C₂²×2_-⁽¹⁺⁴⁾

Subgroups: 3036 in 2936 conjugacy classes, 2836 normal (4 characteristic)
C₁, C₂, C₂^[×6], C₂^[×20], C₄^[×40], C₂²^[×27], C₂²^[×60], C₂×C₄^[×300], D₄^[×160], Q₈^[×160], C₂³, C₂³^[×30], C₂³^[×20], C₂²×C₄^[×190], C₂×D₄^[×120], C₂×Q₈^[×440], C₄○D₄^[×640], C₂⁴^[×5], C₂³×C₄^[×15], C₂²×D₄^[×10], C₂²×Q₈^[×130], C₂×C₄○D₄^[×240], 2_-⁽¹⁺⁴⁾^[×256], Q₈×C₂³^[×5], C₂²×C₄○D₄^[×10], C₂×2_-⁽¹⁺⁴⁾^[×48], C₂²×2_-⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×63], C₂²^[×651], C₂³^[×1395], C₂⁴^[×651], 2_-⁽¹⁺⁴⁾^[×4], C₂⁵^[×63], C₂×2_-⁽¹⁺⁴⁾^[×6], C₂⁶, C₂²×2_-⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c⁴=d²=1, e²=f²=c², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef^-1=c²e >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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

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

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

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

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

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

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,2,4,0,0,0,2,0,0,0,0,2,3,0,4,0,0,0,4,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,0,4,0,1,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,2,4,0,0,1,0,1,2,0,0,0,0,4,4,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,1,4,0,0,3,0,3,1,0,0,0,0,2,0,0,0,0,0,1,3] >;

68 conjugacy classes

class	1	2A	···	2G	2H	···	2AA	4A	···	4AN
order	1	2	···	2	2	···	2	4	···	4
size	1	1	···	1	2	···	2	2	···	2

68 irreducible representations

dim	1	1	1	1	4
type	+	+	+	+	-
image	C₁	C₂	C₂	C₂	2_-⁽¹⁺⁴⁾
kernel	C₂²×2_-⁽¹⁺⁴⁾	Q₈×C₂³	C₂²×C₄○D₄	C₂×2_-⁽¹⁺⁴⁾	C₂²
# reps	1	5	10	48	4

In GAP, Magma, Sage, TeX

C_2^2\times 2_-^{(1+4)}

% in TeX

G:=Group("C2^2xES-(2,2)");

// GroupNames label

G:=SmallGroup(128,2324);

// by ID

G=gap.SmallGroup(128,2324);

# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925,456,723,352,2019]);

// Polycyclic

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

// generators/relations

G = C₂²×2_-⁽¹⁺⁴⁾ order 128 = 2⁷

Direct product of C₂² and 2_-⁽¹⁺⁴⁾

G = C22×2- (1+4) order 128 = 27

Direct product of C22 and 2- (1+4)

G = C₂²×2_-⁽¹⁺⁴⁾ order 128 = 2⁷

Direct product of C₂² and 2_-⁽¹⁺⁴⁾