C2^2xC4oD8 - GroupNames

G = C₂²xC₄oD₈ order 128 = 2⁷

Direct product of C₂² and C₄oD₈

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂²xC₄oD₈, D₈:₇C₂³, C₄.₄C₂⁵, C₈.₂₀C₂⁴, Q₁₆:₇C₂³, D₄.₂C₂⁴, Q₈.₂C₂⁴, SD₁₆:₆C₂³, C₂⁴.₁₄₅D₄, D₈ o(C₂²xC₄), C₄ o(C₂²xD₈), Q₁₆ o(C₂²xC₄), C₄ o(C₂²xQ₁₆), (C₂xC₈):₁₅C₂³, (C₂³xC₈):₁₁C₂, C₄oD₄:₄C₂³, C₄ o(C₂²xSD₁₆), SD₁₆ o(C₂²xC₄), (C₂²xD₈):₂₄C₂, (C₂xD₈):₅₉C₂², C₄.₃₀(C₂²xD₄), C₂.₃₉(D₄xC₂³), (C₂xC₄).₆₁₀C₂⁴, (C₂²xC₈):₆₇C₂², (C₂²xQ₁₆):₂₄C₂, (C₂xQ₁₆):₆₃C₂², (C₂²xC₄).₆₃₀D₄, C₂³.₄₁₀(C₂xD₄), C₂².₅(C₂²xD₄), (C₂²xSD₁₆):₃₀C₂, (C₂xSD₁₆):₈₂C₂², (C₂xD₄).₄₉₀C₂³, (C₂xQ₈).₄₇₄C₂³, (C₂³xC₄).₇₁₄C₂², (C₂²xC₄).₁₅₉₂C₂³, (C₂²xD₄).₆₀₃C₂², (C₂²xQ₈).₅₀₄C₂², C₄ o(C₂xC₄oD₈), (C₂xC₄)o₂(C₂xD₈), (C₂xC₄)o(C₄oD₈), (C₂xC₄)o₂(C₂xQ₁₆), (C₂xC₄)o(C₂²xD₈), (C₂²xC₄)o(C₂xD₈), (C₂xC₄)o₂(C₂xSD₁₆), (C₂xC₄)o(C₂²xQ₁₆), (C₂²xC₄)o(C₂xQ₁₆), (C₂xC₄).₈₈₃(C₂xD₄), (C₂²xC₄)o(C₂xSD₁₆), (C₂xC₄)o(C₂²xSD₁₆), (C₂²xC₄)o(C₂²xD₈), (C₂xC₄oD₄):₇₆C₂², (C₂²xC₄oD₄):₂₅C₂, (C₂²xC₄)o(C₂²xQ₁₆), (C₂²xC₄)o(C₂²xSD₁₆), (C₂xC₄)o(C₂xC₄oD₈), SmallGroup(128,2309)

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

Derived series

C₁ — C₄ — C₂²xC₄oD₈

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄oD₄ — C₂²xC₄oD₈

Lower central

C₁ — C₂ — C₄ — C₂²xC₄oD₈

Upper central

C₁ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄oD₈

Jennings

C₁ — C₂ — C₂ — C₄ — C₂²xC₄oD₈

Generators and relations for C₂²xC₄oD₈
G = < a,b,c,d,e | a²=b²=c⁴=e²=1, d⁴=c², ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c²d³ >

Subgroups: 1148 in 752 conjugacy classes, 436 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₂xC₈, D₈, SD₁₆, Q₁₆, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂⁴, C₂²xC₈, C₂²xC₈, C₂xD₈, C₂xSD₁₆, C₂xQ₁₆, C₄oD₈, C₂³xC₄, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, C₂³xC₈, C₂²xD₈, C₂²xSD₁₆, C₂²xQ₁₆, C₂xC₄oD₈, C₂²xC₄oD₄, C₂²xC₄oD₈
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₄oD₈, C₂²xD₄, C₂⁵, C₂xC₄oD₈, D₄xC₂³, C₂²xC₄oD₈

Smallest permutation representation of C₂²xC₄oD₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2S	4A	···	4H	4I	4J	4K	4L	4M	···	4T	8A	···	8P
order	1	2	···	2	2	2	2	2	2	···	2	4	···	4	4	4	4	4	4	···	4	8	···	8
size	1	1	···	1	2	2	2	2	4	···	4	1	···	1	2	2	2	2	4	···	4	2	···	2

56 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	C₄oD₈
kernel	C₂²xC₄oD₈	C₂³xC₈	C₂²xD₈	C₂²xSD₁₆	C₂²xQ₁₆	C₂xC₄oD₈	C₂²xC₄oD₄	C₂²xC₄	C₂⁴	C₂²
# reps	1	1	1	2	1	24	2	7	1	16

Matrix representation of C₂²xC₄oD₈ ►in GL₄(F₁₇) generated by

16	0	0	0
0	16	0	0
0	0	16	0
0	0	0	16

16	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

16	0	0	0
0	16	0	0
0	0	4	0
0	0	0	4

16	0	0	0
0	1	0	0
0	0	3	14
0	0	3	3

16	0	0	0
0	16	0	0
0	0	3	14
0	0	14	14

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[16,0,0,0,0,1,0,0,0,0,3,3,0,0,14,3],[16,0,0,0,0,16,0,0,0,0,3,14,0,0,14,14] >;

C₂²xC₄oD₈ in GAP, Magma, Sage, TeX

C_2^2\times C_4\circ D_8

% in TeX

G:=Group("C2^2xC4oD8");

// GroupNames label

G:=SmallGroup(128,2309);

// by ID

G=gap.SmallGroup(128,2309);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,352,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^4=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,c*e=e*c,e*d*e=c^2*d^3>;

// generators/relations

G = C22xC4oD8 order 128 = 27

Direct product of C22 and C4oD8

G = C₂²xC₄oD₈ order 128 = 2⁷

Direct product of C₂² and C₄oD₈