C2^2xC2^2:Q8

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

Aliases: C₂²×C₂²⋊Q₈, C₂⁴⋊₉Q₈, C₂⁵.₉₅C₂², C₂².₂₂C₂⁵, C₂⁴.₆₀₇C₂³, C₂³.₁₁₁C₂⁴, C₂³⋊₅(C₂×Q₈), C₄⋊C₄⋊₁₄C₂³, (Q₈×C₂³)⋊₇C₂, C₂.₆(D₄×C₂³), C₂.₃(Q₈×C₂³), (C₂⁴×C₄).₁₄C₂, (C₂×C₄).₂₇C₂⁴, (C₂×Q₈)⋊₁₃C₂³, C₂²⋊₁(C₂²×Q₈), C₂³.₈₈₉(C₂×D₄), (C₂²×C₄).₈₀₅D₄, C₄.₁₆₈(C₂²×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

Subgroups: 1324 in 920 conjugacy classes, 516 normal (13 characteristic)
C₁, C₂^[×3], C₂^[×12], C₂^[×8], C₄^[×8], C₄^[×20], C₂², C₂²^[×42], C₂²^[×56], C₂×C₄^[×48], C₂×C₄^[×92], Q₈^[×32], C₂³^[×43], C₂³^[×56], C₂²⋊C₄^[×32], C₄⋊C₄^[×48], C₂²×C₄^[×60], C₂²×C₄^[×68], C₂×Q₈^[×16], C₂×Q₈^[×48], C₂⁴, C₂⁴^[×14], C₂⁴^[×8], C₂×C₂²⋊C₄^[×24], C₂×C₄⋊C₄^[×36], C₂²⋊Q₈^[×64], C₂³×C₄^[×2], C₂³×C₄^[×16], C₂³×C₄^[×8], C₂²×Q₈^[×12], C₂²×Q₈^[×8], C₂⁵, C₂²×C₂²⋊C₄^[×2], C₂²×C₄⋊C₄, C₂²×C₄⋊C₄^[×2], C₂×C₂²⋊Q₈^[×24], C₂⁴×C₄, Q₈×C₂³, C₂²×C₂²⋊Q₈

Quotients:
C₁, C₂^[×31], C₂²^[×155], D₄^[×8], Q₈^[×8], C₂³^[×155], C₂×D₄^[×28], C₂×Q₈^[×28], C₄○D₄^[×4], C₂⁴^[×31], C₂²⋊Q₈^[×16], C₂²×D₄^[×14], C₂²×Q₈^[×14], C₂×C₄○D₄^[×6], C₂⁵, C₂×C₂²⋊Q₈^[×12], D₄×C₂³, Q₈×C₂³, C₂²×C₄○D₄, C₂²×C₂²⋊Q₈

Generators and relations
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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

G = C₂²×C₂²⋊Q₈ order 128 = 2⁷

Direct product of C₂² and C₂²⋊Q₈