C2xC4^2.78C2^2 - GroupNames

G = C₂×C₄².₇₈C₂² order 128 = 2⁷

Direct product of C₂ and C₄².₇₈C₂²

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

Aliases: C₂×C₄².₇₈C₂², C₄².₃₅₄D₄, C₄².₇₀₅C₂³, (C₄×C₈)⋊₇₁C₂², C₄⋊C₄.₈₃C₂³, (C₂×C₈).₄₉₀C₂³, (C₂×C₄).₃₂₈C₂⁴, (C₂×D₄).₉₇C₂³, C₂³.₈₇₂(C₂×D₄), (C₂²×C₄).₅₆₅D₄, (C₂×Q₈).₈₅C₂³, Q₈⋊C₄⋊₆₁C₂², C₄.₂₀(C₄.₄D₄), C₂².₉₇(C₄○D₈), C₄².C₂⋊₃₃C₂², (C₂²×C₈).₅₁₉C₂², C₂².₅₈₈(C₂²×D₄), D₄⋊C₄.₁₄₃C₂², (C₂²×C₄).₁₅₅₀C₂³, (C₂×C₄²).₁₁₂₃C₂², C₄.₄D₄.₁₃₂C₂², C₂².₈₂(C₄.₄D₄), (C₂²×D₄).₃₆₅C₂², (C₂²×Q₈).₂₉₈C₂², (C₂×C₄×C₈)⋊₂₁C₂, C₂.₂₈(C₂×C₄○D₈), C₄.₃₇(C₂×C₄○D₄), (C₂×C₄).₆₉₃(C₂×D₄), (C₂×Q₈⋊C₄)⋊₁₉C₂, (C₂×C₄².C₂)⋊₃₃C₂, C₂.₃₉(C₂×C₄.₄D₄), (C₂×D₄⋊C₄).₁₉C₂, (C₂×C₄).₇₀₇(C₄○D₄), (C₂×C₄⋊C₄).₆₂₀C₂², (C₂×C₄.₄D₄).₃₈C₂, SmallGroup(128,1862)

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

Derived series

C₁ — C₂×C₄ — C₂×C₄².₇₈C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂×C₄×C₈ — C₂×C₄².₇₈C₂²

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₄².₇₈C₂²

Upper central

C₁ — C₂³ — C₂×C₄² — C₂×C₄².₇₈C₂²

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₄².₇₈C₂²

Generators and relations for C₂×C₄².₇₈C₂²
G = < a,b,c,d,e | a²=b⁴=c⁴=d²=1, e²=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b^-1c², be=eb, dcd=c^-1, ce=ec, ede^-1=b²cd >

Subgroups: 404 in 206 conjugacy classes, 100 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₄.₄D₄, C₄².C₂, C₄².C₂, C₂²×C₈, C₂²×D₄, C₂²×Q₈, C₂×C₄×C₈, C₂×D₄⋊C₄, C₂×Q₈⋊C₄, C₄².₇₈C₂², C₂×C₄.₄D₄, C₂×C₄².C₂, C₂×C₄².₇₈C₂²
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄.₄D₄, C₄○D₈, C₂²×D₄, C₂×C₄○D₄, C₄².₇₈C₂², C₂×C₄.₄D₄, C₂×C₄○D₈, C₂×C₄².₇₈C₂²

Smallest permutation representation of C₂×C₄².₇₈C₂²
►On 64 points

Generators in S₆₄

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

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

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

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

(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)

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

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

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

44 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4L	4M	···	4R	8A	···	8P
order	1	2	···	2	2	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	8	8	2	···	2	8	···	8	2	···	2

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	C₄○D₄	C₄○D₈
kernel	C₂×C₄².₇₈C₂²	C₂×C₄×C₈	C₂×D₄⋊C₄	C₂×Q₈⋊C₄	C₄².₇₈C₂²	C₂×C₄.₄D₄	C₂×C₄².C₂	C₄²	C₂²×C₄	C₂×C₄	C₂²
# reps	1	1	2	2	8	1	1	2	2	8	16

Matrix representation of C₂×C₄².₇₈C₂² ►in GL₅(𝔽₁₇)

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

16	0	0	0	0
0	7	8	0	0
0	15	10	0	0
0	0	0	0	13
0	0	0	4	0

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

1	0	0	0	0
0	16	0	0	0
0	6	1	0	0
0	0	0	16	0
0	0	0	0	1

1	0	0	0	0
0	6	2	0	0
0	8	11	0	0
0	0	0	3	14
0	0	0	3	3

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,7,15,0,0,0,8,10,0,0,0,0,0,0,4,0,0,0,13,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[1,0,0,0,0,0,16,6,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,8,0,0,0,2,11,0,0,0,0,0,3,3,0,0,0,14,3] >;

C₂×C₄².₇₈C₂² in GAP, Magma, Sage, TeX

C_2\times C_4^2._{78}C_2^2

% in TeX

G:=Group("C2xC4^2.78C2^2");

// GroupNames label

G:=SmallGroup(128,1862);

// by ID

G=gap.SmallGroup(128,1862);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,2804,172,4037,124]);

// Polycyclic

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

// generators/relations

G = C2×C42.78C22 order 128 = 27

Direct product of C2 and C42.78C22

G = C₂×C₄².₇₈C₂² order 128 = 2⁷

Direct product of C₂ and C₄².₇₈C₂²