C4xC2^2:Q8 - GroupNames

G = C₄xC₂²:Q₈ order 128 = 2⁷

Direct product of C₄ and C₂²:Q₈

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

Aliases: C₄xC₂²:Q₈, C₄².₄₃₈D₄, C₂³.₁₈₄C₂⁴, C₂⁴.₅₃₈C₂³, C₂²:₁(C₄xQ₈), C₄.₁₂₈(C₄xD₄), (C₂²xC₄):₂₀Q₈, C₂³.₉₀(C₂xQ₈), C₂².₇₅(C₂³xC₄), (C₂²xC₄²).₂₅C₂, C₄ o₅(C₂³.₇Q₈), C₄ o₅(C₂³.₈Q₈), C₂².₈₀(C₂²xD₄), C₂³.₂₁₉(C₄oD₄), C₂².₂₇(C₂²xQ₈), (C₂³xC₄).₆₈₀C₂², C₂³.₁₂₁(C₂²xC₄), (C₂xC₄²).₄₀₃C₂², (C₂²xC₄).₇₄₈C₂³, C₂³.₈Q₈.₇₃C₂, C₂³.₇Q₈.₈₃C₂, C₂.₈(C₂².₁₉C₂⁴), (C₂²xQ₈).₃₉₂C₂², C₄ o₃(C₂³.₆₇C₂³), C₄ o₅(C₂³.₆₃C₂³), C₄ o₅(C₂³.₆₅C₂³), C₂³.₆₇C₂³:₁₁₃C₂, C₂³.₆₃C₂³:₂₁₁C₂, C₂³.₆₅C₂³:₁₇₅C₂, C₂.C₄².₅₁₇C₂², C₂.₄(C₂³.₃₇C₂³), C₂.₈(C₂³.₃₆C₂³), (C₂xC₄xQ₈):₂C₂, C₂.₈(C₂xC₄xQ₈), (C₄xC₄:C₄):₂₁C₂, C₂.₁₆(C₂xC₄xD₄), C₄:C₄:₂₆(C₂xC₄), (C₂xQ₈):₂₂(C₂xC₄), C₂.₁₃(C₄xC₄oD₄), C₂.₆(C₂xC₂²:Q₈), (C₂xC₄).₃₅₃(C₂xQ₈), (C₂xC₄).₁₃₉₁(C₂xD₄), (C₄xC₂²:C₄).₁₉C₂, C₂²:C₄.₂₉(C₂xC₄), (C₂xC₄).₂₀(C₂²xC₄), C₂².₇₆(C₂xC₄oD₄), (C₂xC₂²:Q₈).₆₂C₂, (C₂xC₄).₆₃₈(C₄oD₄), (C₂xC₄:C₄).₇₉₈C₂², (C₂²xC₄).₄₁₂(C₂xC₄), (C₂xC₄)o₃(C₂³.₇Q₈), (C₂xC₂²:C₄).₄₇₉C₂², (C₂xC₄)o₃(C₂³.₆₅C₂³), (C₂xC₄)o(C₂xC₂²:Q₈), SmallGroup(128,1034)

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

Derived series

C₁ — C₂² — C₄xC₂²:Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄² — C₄xC₂²:Q₈

Lower central

C₁ — C₂² — C₄xC₂²:Q₈

Upper central

C₁ — C₂²xC₄ — C₄xC₂²:Q₈

Jennings

C₁ — C₂³ — C₄xC₂²:Q₈

Generators and relations for C₄xC₂²:Q₈
G = < a,b,c,d,e | a⁴=b²=c²=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 492 in 328 conjugacy classes, 172 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₂⁴, C₂.C₄², C₂xC₄², C₂xC₄², C₂xC₄², C₂xC₂²:C₄, C₂xC₄:C₄, C₂xC₄:C₄, C₄xQ₈, C₂²:Q₈, C₂³xC₄, C₂²xQ₈, C₄xC₂²:C₄, C₄xC₄:C₄, C₄xC₄:C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂²xC₄², C₂xC₄xQ₈, C₂xC₂²:Q₈, C₄xC₂²:Q₈
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, Q₈, C₂³, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₄xD₄, C₄xQ₈, C₂²:Q₈, C₂³xC₄, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄xD₄, C₂xC₄xQ₈, C₄xC₄oD₄, C₂xC₂²:Q₈, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₄xC₂²:Q₈

Smallest permutation representation of C₄xC₂²:Q₈
►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 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 27)(18 28)(19 25)(20 26)(29 44)(30 41)(31 42)(32 43)(33 40)(34 37)(35 38)(36 39)(45 50)(46 51)(47 52)(48 49)(53 60)(54 57)(55 58)(56 59)

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4AB	4AC	···	4AR
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	···	4
size	1	1	···	1	2	2	2	2	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	C₄oD₄	C₄oD₄
kernel	C₄xC₂²:Q₈	C₄xC₂²:C₄	C₄xC₄:C₄	C₂³.₇Q₈	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂²xC₄²	C₂xC₄xQ₈	C₂xC₂²:Q₈	C₂²:Q₈	C₄²	C₂²xC₄	C₂xC₄	C₂³
# reps	1	2	3	1	2	2	1	1	1	1	1	16	4	4	12	4

Matrix representation of C₄xC₂²:Q₈ ►in GL₆(F₅)

3	0	0	0	0	0
0	3	0	0	0	0
0	0	1	0	0	0
0	0	0	1	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	1

1	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	4	0
0	0	0	0	0	4

2	0	0	0	0	0
0	3	0	0	0	0
0	0	3	0	0	0
0	0	0	2	0	0
0	0	0	0	3	0
0	0	0	0	0	2

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

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

C₄xC₂²:Q₈ in GAP, Magma, Sage, TeX

C_4\times C_2^2\rtimes Q_8

% in TeX

G:=Group("C4xC2^2:Q8");

// GroupNames label

G:=SmallGroup(128,1034);

// by ID

G=gap.SmallGroup(128,1034);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,304]);

// Polycyclic

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

// generators/relations

G = C4xC22:Q8 order 128 = 27

Direct product of C4 and C22:Q8

G = C₄xC₂²:Q₈ order 128 = 2⁷

Direct product of C₄ and C₂²:Q₈