C2xC2^3.11D4

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

Aliases: C₂×C₂³.₁₁D₄, C₂⁴.₉₃D₄, C₂⁵.₃₀C₂², C₂³.₂₉₀C₂⁴, C₂⁴.₆₄₉C₂³, C₂³.₁₄₄(C₂×D₄), (C₂²×C₄).₃₆₆D₄, C₂³.₃₆₉(C₄○D₄), (C₂³×C₄).₃₂₁C₂², (C₂²×C₄).₄₉₅C₂³, C₂².₁₇₃(C₂²×D₄), C₂².₁₆₃(C₄⋊D₄), C₂.C₄²⋊₅₄C₂², C₂².₇₈(C₄.₄D₄), C₂².₃₃(C₄²⋊₂C₂), C₂².₁₀₂(C₂².D₄), (C₂²×C₄⋊C₄)⋊₁₄C₂, (C₂×C₄).₂₉₁(C₂×D₄), C₂.₁₀(C₂×C₄⋊D₄), C₂.₈(C₂×C₄.₄D₄), (C₂×C₄⋊C₄)⋊₁₀₇C₂², C₂.₆(C₂×C₄²⋊₂C₂), C₂².₁₇₀(C₂×C₄○D₄), C₂.₈(C₂×C₂².D₄), (C₂×C₂.C₄²)⋊₂₆C₂, (C₂²×C₂²⋊C₄).₂₀C₂, (C₂×C₂²⋊C₄).₄₈₅C₂², SmallGroup(128,1122)

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

Derived series

C₁ — C₂³ — C₂×C₂³.₁₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₂²⋊C₄ — C₂×C₂³.₁₁D₄

Lower central

C₁ — C₂³ — C₂×C₂³.₁₁D₄

Upper central

C₁ — C₂⁴ — C₂×C₂³.₁₁D₄

Jennings

C₁ — C₂³ — C₂×C₂³.₁₁D₄

Subgroups: 820 in 410 conjugacy classes, 148 normal (16 characteristic)
C₁, C₂^[×3], C₂^[×12], C₂^[×4], C₄^[×14], C₂²^[×3], C₂²^[×32], C₂²^[×36], C₂×C₄^[×4], C₂×C₄^[×62], C₂³, C₂³^[×18], C₂³^[×52], C₂²⋊C₄^[×24], C₄⋊C₄^[×8], C₂²×C₄^[×16], C₂²×C₄^[×34], C₂⁴, C₂⁴^[×6], C₂⁴^[×12], C₂.C₄²^[×12], C₂×C₂²⋊C₄^[×12], C₂×C₂²⋊C₄^[×12], C₂×C₄⋊C₄^[×4], C₂×C₄⋊C₄^[×4], C₂³×C₄^[×2], C₂³×C₄^[×4], C₂⁵, C₂×C₂.C₄², C₂×C₂.C₄²^[×2], C₂³.₁₁D₄^[×8], C₂²×C₂²⋊C₄, C₂²×C₂²⋊C₄^[×2], C₂²×C₄⋊C₄, C₂×C₂³.₁₁D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×8], C₂³^[×15], C₂×D₄^[×12], C₄○D₄^[×10], C₂⁴, C₄⋊D₄^[×4], C₂².D₄^[×12], C₄.₄D₄^[×4], C₄²⋊₂C₂^[×8], C₂²×D₄^[×2], C₂×C₄○D₄^[×5], C₂³.₁₁D₄^[×8], C₂×C₄⋊D₄, C₂×C₂².D₄^[×3], C₂×C₄.₄D₄, C₂×C₄²⋊₂C₂^[×2], C₂×C₂³.₁₁D₄

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=bc=cb, bd=db, fbf^-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₇(𝔽₅)

4	0	0	0	0	0	0
0	4	0	0	0	0	0
0	0	4	0	0	0	0
0	0	0	4	0	0	0
0	0	0	0	4	0	0
0	0	0	0	0	4	0
0	0	0	0	0	0	4

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

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

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

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

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,3,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,0,4,4,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,3,0] >;

44 conjugacy classes

class	1	2A	···	2O	2P	2Q	2R	2S	4A	···	4X
order	1	2	···	2	2	2	2	2	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	C₄○D₄
kernel	C₂×C₂³.₁₁D₄	C₂×C₂.C₄²	C₂³.₁₁D₄	C₂²×C₂²⋊C₄	C₂²×C₄⋊C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	3	8	3	1	4	4	20

In GAP, Magma, Sage, TeX

C_2\times C_2^3._{11}D_4

% in TeX

G:=Group("C2xC2^3.11D4");

// GroupNames label

G:=SmallGroup(128,1122);

// by ID

G=gap.SmallGroup(128,1122);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100]);

// Polycyclic

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

// generators/relations

G = C₂×C₂³.₁₁D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₁₁D₄

G = C2×C23.11D4 order 128 = 27

Direct product of C2 and C23.11D4

G = C₂×C₂³.₁₁D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₁₁D₄