C4^2.207C2^3 - GroupNames

G = C₄².₂₀₇C₂³ order 128 = 2⁷

68^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₂₀₇C₂³, D₄⋊C₈⋊₂₂C₂, C₄⋊C₄.₂₉₆D₄, D₄⋊Q₈⋊₃C₂, (C₂×D₄).₅₂D₄, (C₂×Q₈).₂₆D₄, C₄.₃₇(C₄○D₈), (C₄×C₈).₂₈C₂², C₄.₆Q₁₆⋊₈C₂, C₄.SD₁₆⋊₆C₂, C₄⋊Q₈.₂₇C₂², D₄.D₄⋊₃₂C₂, C₄⋊C₈.₁₆₈C₂², C₄.₃₈(C₈⋊C₂²), (C₄×D₄).₃₆C₂², C₂.₂₂(D₄⋊D₄), C₄.₃₇(C₈.C₂²), C₂.₂₂(D₄.₇D₄), C₂.₁₉(D₄.₉D₄), C₂².₁₇₃C₂²≀C₂, C₂².₄₉C₂⁴.₂C₂, (C₂×C₄).₉₆₄(C₂×D₄), SmallGroup(128,378)

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

Derived series

C₁ — C₄² — C₄².₂₀₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₂².₄₉C₂⁴ — C₄².₂₀₇C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₂₀₇C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₂₀₇C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₂₀₇C₂³

Generators and relations for C₄².₂₀₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=b², d²=a², e²=a²b², ab=ba, cac^-1=dad^-1=a^-1, eae^-1=ab², cbc^-1=dbd^-1=ebe^-1=b^-1, dcd^-1=ac, ece^-1=bc, de=ed >

Subgroups: 256 in 106 conjugacy classes, 34 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊Q₈, C₂×SD₁₆, D₄⋊C₈, C₄.₆Q₁₆, D₄.D₄, D₄⋊Q₈, C₄.SD₁₆, C₂².₄₉C₂⁴, C₄².₂₀₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈⋊C₂², C₈.C₂², D₄⋊D₄, D₄.₇D₄, D₄.₉D₄, C₄².₂₀₇C₂³

Character table of C₄².₂₀₇C₂³

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	8	8	2	2	2	2	4	4	4	4	4	8	8	16	4	4	4	4	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	0	2	-2	2	-2	0	-2	0	-2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	2	-2	2	-2	2	-2	0	-2	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	-2	-2	2	-2	2	2	0	2	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	2	-2	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	0	2	-2	2	-2	0	2	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	0	2	0	-2	0	0	2i	0	-2i	0	0	0	0	√2	√2	-√2	-√2	√-2	0	0	-√-2	complex lifted from C₄○D₈
ρ₁₆	2	-2	2	-2	0	0	0	2	0	-2	-2i	0	2i	0	0	0	0	0	-√-2	√-2	-√-2	√-2	0	√2	-√2	0	complex lifted from C₄○D₈
ρ₁₇	2	2	-2	-2	0	0	2	0	-2	0	0	-2i	0	2i	0	0	0	0	√2	√2	-√2	-√2	-√-2	0	0	√-2	complex lifted from C₄○D₈
ρ₁₈	2	-2	2	-2	0	0	0	2	0	-2	2i	0	-2i	0	0	0	0	0	-√-2	√-2	-√-2	√-2	0	-√2	√2	0	complex lifted from C₄○D₈
ρ₁₉	2	2	-2	-2	0	0	2	0	-2	0	0	-2i	0	2i	0	0	0	0	-√2	-√2	√2	√2	√-2	0	0	-√-2	complex lifted from C₄○D₈
ρ₂₀	2	-2	2	-2	0	0	0	2	0	-2	2i	0	-2i	0	0	0	0	0	√-2	-√-2	√-2	-√-2	0	√2	-√2	0	complex lifted from C₄○D₈
ρ₂₁	2	2	-2	-2	0	0	2	0	-2	0	0	2i	0	-2i	0	0	0	0	-√2	-√2	√2	√2	-√-2	0	0	√-2	complex lifted from C₄○D₈
ρ₂₂	2	-2	2	-2	0	0	0	2	0	-2	-2i	0	2i	0	0	0	0	0	√-2	-√-2	√-2	-√-2	0	-√2	√2	0	complex lifted from C₄○D₈
ρ₂₃	4	-4	4	-4	0	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₄	4	4	-4	-4	0	0	-4	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₅	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	-2i	2i	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₆	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	2i	-2i	0	0	0	0	complex lifted from D₄.₉D₄

Smallest permutation representation of C₄².₂₀₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

Matrix representation of C₄².₂₀₇C₂³ ►in GL₄(𝔽₁₇) generated by

4	0	0	0
0	13	0	0
0	0	0	1
0	0	16	0

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

0	8	0	0
15	0	0	0
0	0	12	12
0	0	12	5

0	1	0	0
16	0	0	0
0	0	0	4
0	0	4	0

13	0	0	0
0	13	0	0
0	0	0	16
0	0	16	0

G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[0,15,0,0,8,0,0,0,0,0,12,12,0,0,12,5],[0,16,0,0,1,0,0,0,0,0,0,4,0,0,4,0],[13,0,0,0,0,13,0,0,0,0,0,16,0,0,16,0] >;

C₄².₂₀₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{207}C_2^3

% in TeX

G:=Group("C4^2.207C2^3");

// GroupNames label

G:=SmallGroup(128,378);

// by ID

G=gap.SmallGroup(128,378);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,520,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₂₀₇C₂³ in TeX

G = C42.207C23 order 128 = 27

68th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₀₇C₂³ order 128 = 2⁷

68^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²