(C2xC8).55D4 - GroupNames

G = (C₂×C₈).₅₅D₄ order 128 = 2⁷

23^rd non-split extension by C₂×C₈ of D₄ acting via D₄/C₂=C₂²

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

Aliases: (C₂×C₈).₅₅D₄, (C₂×D₄).₁₁₉D₄, (C₂×Q₈).₁₁₀D₄, C₄.C₄²⋊₁₃C₂, C₄.₅₇(C₄.₄D₄), C₂².C₄²⋊₂₄C₂, C₂.₂₂(D₄.₄D₄), C₂.₂₂(D₄.₅D₄), C₂³.₂₈₂(C₄○D₄), (C₂²×C₄).₇₃₇C₂³, (C₂²×C₈).₁₁₇C₂², C₂².₂₅₄(C₄⋊D₄), C₂³.₃₆D₄.₁₂C₂, C₂².₁₆(C₄²⋊₂C₂), C₄.₁₁₃(C₂².D₄), C₂.₁₃(C₂³.₁₁D₄), (C₂×M₄(2)).₂₃₅C₂², (C₂×C₂.D₈)⋊₁₂C₂, (C₂×C₄).₈₄(C₄○D₄), (C₂×C₄).₁₃₇₆(C₂×D₄), (C₂×C₄⋊C₄).₁₅₂C₂², (C₂²×C₈)⋊C₂.₆C₂, (C₂×C₄○D₄).₆₇C₂², SmallGroup(128,810)

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

Derived series

C₁ — C₂²×C₄ — (C₂×C₈).₅₅D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₃₆D₄ — (C₂×C₈).₅₅D₄

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×C₈).₅₅D₄

Upper central

C₁ — C₂² — C₂²×C₄ — (C₂×C₈).₅₅D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂×C₈).₅₅D₄

Generators and relations for (C₂×C₈).₅₅D₄
G = < a,b,c,d | a²=b⁸=d²=1, c⁴=b⁴, dbd=ab=ba, ac=ca, ad=da, cbc^-1=ab^-1, dcd=ab⁴c³ >

Subgroups: 216 in 99 conjugacy classes, 38 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₂×C₄⋊C₄, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×C₄○D₄, C₄.C₄², C₂².C₄², (C₂²×C₈)⋊C₂, C₂³.₃₆D₄, C₂×C₂.D₈, (C₂×C₈).₅₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂³.₁₁D₄, D₄.₄D₄, D₄.₅D₄, (C₂×C₈).₅₅D₄

Character table of (C₂×C₈).₅₅D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	8	2	2	2	2	8	8	8	8	8	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	-1	1	1	1	-1	-1	-1	-1	-1	-1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	-1	1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	1	-1	1	1	-1	1	linear of order 2
ρ₄	1	1	1	1	1	1	-1	1	1	1	1	-1	-1	-1	-1	-1	1	1	1	1	-1	1	1	1	1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	-1	1	-1	-1	1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	1	-1	1	1	1	1	1	-1	-1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	1	1	-1	1	1	1	1	1	1	1	-1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	-1	1	1	1	1	-1	1	1	-1	-1	-1	-1	-1	-1	1	1	-1	-1	1	1	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	2	2	-2	-2	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	0	2	2	-2	-2	0	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	-2	-2	-2	2	2	0	0	0	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	2	-2	-2	2	2	0	0	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	-2	2	2	-2	0	-2	2	-2	2	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₄	2	-2	-2	2	-2	2	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	-2	2	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	complex lifted from C₄○D₄
ρ₁₆	2	-2	-2	2	2	-2	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	2i	0	0	-2i	0	complex lifted from C₄○D₄
ρ₁₇	2	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	-2i	0	0	0	0	2i	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	2	-2	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	-2i	0	0	2i	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	-2	2	-2	2	0	-2	2	2	-2	2i	0	0	0	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	2	-2	0	-2	2	-2	2	0	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	2i	0	0	0	0	-2i	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	-2	2	0	-2	2	2	-2	-2i	0	0	0	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₄	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₅	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	2√2	0	0	-2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₆	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	-2√2	0	0	2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2

Smallest permutation representation of (C₂×C₈).₅₅D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

Matrix representation of (C₂×C₈).₅₅D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

5	16	0	0	0	0
9	12	0	0	0	0
0	0	3	6	3	6
0	0	14	14	14	14
0	0	14	11	3	6
0	0	3	3	14	14

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	7	0	0
0	0	5	10	0	0
0	0	0	0	0	10
0	0	0	0	12	7

1	0	0	0	0	0
10	16	0	0	0	0
0	0	16	15	0	0
0	0	0	1	0	0
0	0	0	0	16	15
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,9,0,0,0,0,16,12,0,0,0,0,0,0,3,14,14,3,0,0,6,14,11,3,0,0,3,14,3,14,0,0,6,14,6,14],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,0,0,0,7,10,0,0,0,0,0,0,0,12,0,0,0,0,10,7],[1,10,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;

(C₂×C₈).₅₅D₄ in GAP, Magma, Sage, TeX

(C_2\times C_8)._{55}D_4

% in TeX

G:=Group("(C2xC8).55D4");

// GroupNames label

G:=SmallGroup(128,810);

// by ID

G=gap.SmallGroup(128,810);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,718,172,2028,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d=a*b^4*c^3>;

// generators/relations

Export

Character table of (C₂×C₈).₅₅D₄ in TeX

G = (C2×C8).55D4 order 128 = 27

23rd non-split extension by C2×C8 of D4 acting via D4/C2=C22

G = (C₂×C₈).₅₅D₄ order 128 = 2⁷

23^rd non-split extension by C₂×C₈ of D₄ acting via D₄/C₂=C₂²