C4xES-(2,2) - GroupNames

G = C₄×2_-¹⁺⁴ order 128 = 2⁷

Direct product of C₄ and 2_-¹⁺⁴

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

Aliases: C₄×2_-¹⁺⁴, C₂².₁₆C₂⁵, C₄².₅₃₇C₂³, C₂³.₁₁₀C₂⁴, Q₈ ○(C₄×D₄), D₄ ○(C₄×Q₈), C₂.₁₂(C₂⁴×C₄), C₄.₄₂(C₂³×C₄), C₄⋊C₄.₅₁₇C₂³, (C₂×C₄).₁₆₂C₂⁴, D₄.₂₅(C₂²×C₄), C₂².₆(C₂³×C₄), Q₈.₂₆(C₂²×C₄), (C₄×D₄).₃₄₈C₂², (C₂×D₄).₅₀₀C₂³, (C₂×Q₈).₄₈₃C₂³, (C₄×Q₈).₃₁₈C₂², C₂.₃(C₂×2_-¹⁺⁴), C₂²⋊C₄.₁₂₉C₂³, C₂.₃(C₂.C₂⁵), (C₂×C₄²).₉₁₄C₂², (C₂×2_-¹⁺⁴).₈C₂, (C₂²×C₄).₁₂₉₆C₂³, (C₂²×Q₈).₄₈₃C₂², C₄ ○₂(C₂³.₃₂C₂³), C₄ ○₃(C₂³.₃₃C₂³), C₂³.₃₃C₂³⋊₃₅C₂, C₂³.₃₂C₂³⋊₂₁C₂, C₄²⋊C₂.₃₃₆C₂², C₄⋊C₄ ○(C₄×Q₈), (C₄×D₄)○(C₄×Q₈), (C₂×D₄)○(C₄×Q₈), (C₂×C₄×Q₈)⋊₄₃C₂, C₂²⋊C₄ ○(C₄×Q₈), C₄○D₄⋊₁₁(C₂×C₄), (C₄×C₄○D₄)⋊₁₅C₂, (C₂×Q₈)⋊₃₀(C₂×C₄), (C₂×C₄).₉₁(C₂²×C₄), (C₂×C₄⋊C₄).₉₄₂C₂², (C₂×C₄○D₄).₃₁₉C₂², SmallGroup(128,2162)

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

Derived series

C₁ — C₂ — C₄×2_-¹⁺⁴

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₂×C₄×Q₈ — C₄×2_-¹⁺⁴

Lower central

C₁ — C₂ — C₄×2_-¹⁺⁴

Upper central

C₁ — C₂×C₄ — C₄×2_-¹⁺⁴

Jennings

C₁ — C₂² — C₄×2_-¹⁺⁴

Generators and relations for C₄×2_-¹⁺⁴
G = < a,b,c,d,e | a⁴=b⁴=c²=1, d²=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=b²d >

Subgroups: 836 in 746 conjugacy classes, 686 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₄×Q₈, C₄×C₄○D₄, C₂³.₃₂C₂³, C₂³.₃₃C₂³, C₂×2_-¹⁺⁴, C₄×2_-¹⁺⁴
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₂³×C₄, 2_-¹⁺⁴, C₂⁵, C₂⁴×C₄, C₂×2_-¹⁺⁴, C₂.C₂⁵, C₄×2_-¹⁺⁴

Smallest permutation representation of C₄×2_-¹⁺⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	4A	4B	4C	4D	4E	···	4BB
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4
size	1	1	1	1	2	···	2	1	1	1	1	2	···	2

68 irreducible representations

dim	1	1	1	1	1	1	1	4	4
type	+	+	+	+	+	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	2_-¹⁺⁴	C₂.C₂⁵
kernel	C₄×2_-¹⁺⁴	C₂×C₄×Q₈	C₄×C₄○D₄	C₂³.₃₂C₂³	C₂³.₃₃C₂³	C₂×2_-¹⁺⁴	2_-¹⁺⁴	C₄	C₂
# reps	1	5	10	5	10	1	32	2	2

Matrix representation of C₄×2_-¹⁺⁴ ►in GL₅(𝔽₅)

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

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

1	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	0	1
0	0	0	1	0

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

4	0	0	0	0
0	0	0	3	0
0	0	0	0	3
0	3	0	0	0
0	0	3	0	0

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

C₄×2_-¹⁺⁴ in GAP, Magma, Sage, TeX

C_4\times 2_-^{1+4}

% in TeX

G:=Group("C4xES-(2,2)");

// GroupNames label

G:=SmallGroup(128,2162);

// by ID

G=gap.SmallGroup(128,2162);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123,172]);

// Polycyclic

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

// generators/relations

G = C4×2- 1+4 order 128 = 27

Direct product of C4 and 2- 1+4

G = C₄×2_-¹⁺⁴ order 128 = 2⁷

Direct product of C₄ and 2_-¹⁺⁴