C4:ES-(2,2) - GroupNames

G = C₄⋊2_-¹⁺⁴ order 128 = 2⁷

The semidirect product of C₄ and 2_-¹⁺⁴ acting via 2_-¹⁺⁴/C₂×Q₈=C₂

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

Aliases: C₄⋊₁2_-¹⁺⁴, C₂³.₄₂C₂⁴, C₂².₈₆C₂⁵, C₄².₅₇₄C₂³, (C₂×Q₈)⋊₂₉D₄, (D₄×Q₈)⋊₁₇C₂, Q₈ ○₂(C₄⋊D₄), Q₈.₆₀(C₂×D₄), Q₈⋊₅D₄⋊₁₆C₂, Q₈⋊₆D₄⋊₁₇C₂, C₂.₃₂(D₄×C₂³), C₄⋊C₄.₄₈₈C₂³, (C₂×C₄).₆₀₃C₂⁴, C₄.₁₂₁(C₂²×D₄), C₄⋊Q₈.₃₃₉C₂², C₂².₄(C₂²×D₄), (C₄×D₄).₂₃₀C₂², (C₂×D₄).₃₀₂C₂³, C₂²⋊C₄.₂₁C₂³, (C₂×2_-¹⁺⁴)⋊₈C₂, (C₄×Q₈).₃₂₆C₂², (C₂×Q₈).₄₄₆C₂³, C₄⋊₁D₄.₁₈₅C₂², C₄⋊D₄.₁₁₄C₂², (C₂²×C₄).₃₅₉C₂³, (C₂×C₄²).₉₄₂C₂², C₂²⋊Q₈.₁₁₃C₂², C₂.₂₂(C₂×2_-¹⁺⁴), C₂.₂₃(C₂.C₂⁵), C₂².₂₆C₂⁴⋊₃₅C₂, C₄.₄D₄.₁₇₄C₂², (C₂²×Q₈).₃₅₈C₂², C₂².₃₁C₂⁴⋊₁₆C₂, (C₂×C₄×Q₈)⋊₅₉C₂, (C₂×Q₈)○(C₄⋊D₄), (C₂×C₄).₁₈₇(C₂×D₄), (C₂×C₄⋊C₄).₉₆₃C₂², (C₂×C₄○D₄).₂₂₇C₂², SmallGroup(128,2229)

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

Derived series

C₁ — C₂² — C₄⋊2_-¹⁺⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂×C₄×Q₈ — C₄⋊2_-¹⁺⁴

Lower central

C₁ — C₂² — C₄⋊2_-¹⁺⁴

Upper central

C₁ — C₂² — 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, bab^-1=a^-1, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=b²d >

Subgroups: 1108 in 746 conjugacy classes, 430 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄⋊C₄, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₄×Q₈, C₂².₂₆C₂⁴, C₂².₃₁C₂⁴, Q₈⋊₅D₄, D₄×Q₈, Q₈⋊₆D₄, C₂×2_-¹⁺⁴, C₄⋊2_-¹⁺⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2_-¹⁺⁴, C₂⁵, D₄×C₂³, 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 49 27 63)(2 52 28 62)(3 51 25 61)(4 50 26 64)(5 60 30 56)(6 59 31 55)(7 58 32 54)(8 57 29 53)(9 19 13 40)(10 18 14 39)(11 17 15 38)(12 20 16 37)(21 41 33 45)(22 44 34 48)(23 43 35 47)(24 42 36 46)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2M	4A	···	4P	4Q	···	4AD
order	1	2	2	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	2_-¹⁺⁴	C₂.C₂⁵
kernel	C₄⋊2_-¹⁺⁴	C₂×C₄×Q₈	C₂².₂₆C₂⁴	C₂².₃₁C₂⁴	Q₈⋊₅D₄	D₄×Q₈	Q₈⋊₆D₄	C₂×2_-¹⁺⁴	C₂×Q₈	C₄	C₂
# reps	1	1	6	6	8	4	4	2	8	2	2

Matrix representation of C₄⋊2_-¹⁺⁴ ►in GL₆(𝔽₅)

0	1	0	0	0	0
4	0	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

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

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

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

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

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

C₄⋊2_-¹⁺⁴ in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("C4:ES-(2,2)");

// GroupNames label

G:=SmallGroup(128,2229);

// by ID

G=gap.SmallGroup(128,2229);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,352,570,136,1684]);

// Polycyclic

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

The semidirect product of C4 and 2- 1+4 acting via 2- 1+4/C2×Q8=C2

G = C₄⋊2_-¹⁺⁴ order 128 = 2⁷

The semidirect product of C₄ and 2_-¹⁺⁴ acting via 2_-¹⁺⁴/C₂×Q₈=C₂