C2^2xC2^2:Q8 - GroupNames

G = C₂²×C₂²⋊Q₈ order 128 = 2⁷

Direct product of C₂² and C₂²⋊Q₈

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

Aliases: C₂²×C₂²⋊Q₈, C₂⁴⋊₉Q₈, C₂⁵.₉₅C₂², C₂².₂₂C₂⁵, C₂³.₁₁₁C₂⁴, C₂⁴.₆₀₇C₂³, C₂³⋊₅(C₂×Q₈), C₄⋊C₄⋊₁₄C₂³, (Q₈×C₂³)⋊₇C₂, C₂.₆(D₄×C₂³), C₂.₃(Q₈×C₂³), (C₂×C₄).₂₇C₂⁴, (C₂⁴×C₄).₁₄C₂, (C₂×Q₈)⋊₁₃C₂³, C₂²⋊₁(C₂²×Q₈), C₄.₁₆₈(C₂²×D₄), (C₂²×C₄).₈₀₅D₄, C₂³.₈₈₉(C₂×D₄), C₂²⋊C₄.₆₉C₂³, (C₂²×Q₈)⋊₅₅C₂², C₂³.₃₇₈(C₄○D₄), (C₂³×C₄).₅₇₈C₂², C₂².₁₅₇(C₂²×D₄), (C₂²×C₄).₁₁₇₁C₂³, (C₂²×C₄⋊C₄)⋊₃₈C₂, C₂.₆(C₂²×C₄○D₄), (C₂×C₄⋊C₄)⋊₁₂₃C₂², (C₂×C₄).₁₄₃₉(C₂×D₄), C₂².₁₄₇(C₂×C₄○D₄), (C₂²×C₂²⋊C₄).₂₇C₂, (C₂×C₂²⋊C₄).₅₂₄C₂², SmallGroup(128,2165)

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

Derived series

C₁ — C₂² — C₂²×C₂²⋊Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴×C₄ — C₂²×C₂²⋊Q₈

Lower central

C₁ — C₂² — C₂²×C₂²⋊Q₈

Upper central

C₁ — C₂⁴ — C₂²×C₂²⋊Q₈

Jennings

C₁ — C₂² — C₂²×C₂²⋊Q₈

Generators and relations for C₂²×C₂²⋊Q₈
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 1324 in 920 conjugacy classes, 516 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂⁴, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂⁵, C₂²×C₂²⋊C₄, C₂²×C₄⋊C₄, C₂²×C₄⋊C₄, C₂×C₂²⋊Q₈, C₂⁴×C₄, Q₈×C₂³, C₂²×C₂²⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂⁵, C₂×C₂²⋊Q₈, D₄×C₂³, Q₈×C₂³, C₂²×C₄○D₄, C₂²×C₂²⋊Q₈

Smallest permutation representation of C₂²×C₂²⋊Q₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2W	4A	···	4P	4Q	···	4AF
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄
kernel	C₂²×C₂²⋊Q₈	C₂²×C₂²⋊C₄	C₂²×C₄⋊C₄	C₂×C₂²⋊Q₈	C₂⁴×C₄	Q₈×C₂³	C₂²×C₄	C₂⁴	C₂³
# reps	1	2	3	24	1	1	8	8	8

Matrix representation of C₂²×C₂²⋊Q₈ ►in GL₆(𝔽₅)

4	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

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

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	0	0	0	1

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

1	0	0	0	0	0
0	4	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

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

C₂²×C₂²⋊Q₈ in GAP, Magma, Sage, TeX

C_2^2\times C_2^2\rtimes Q_8

% in TeX

G:=Group("C2^2xC2^2:Q8");

// GroupNames label

G:=SmallGroup(128,2165);

// by ID

G=gap.SmallGroup(128,2165);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430]);

// Polycyclic

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

// generators/relations

G = C22×C22⋊Q8 order 128 = 27

Direct product of C22 and C22⋊Q8

G = C₂²×C₂²⋊Q₈ order 128 = 2⁷

Direct product of C₂² and C₂²⋊Q₈