C4^2.10C2^3 - GroupNames

G = C₄².₁₀C₂³ order 128 = 2⁷

10^th non-split extension by C₄² of C₂³ acting faithfully

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

Aliases: C₄².₁₀C₂³, C₈⋊Q₈⋊₃C₂, C₄⋊C₄.₄₅D₄, (C₂×D₄).₃₄D₄, (C₂×Q₈).₃₄D₄, C₄⋊Q₈.₄₀C₂², C₈⋊C₄.₉₆C₂², C₂².₁₉₁C₂²≀C₂, C₂.₂₃(D₄.₉D₄), C₂.₂₅(D₄.₈D₄), C₄².C₂.₈C₂², C₄².₂C₂²⋊₅C₂, C₄.₄D₄.₁₃C₂², C₂.₂₃(D₄.₁₀D₄), C₄².C₂².₂C₂, C₄².₃₀C₂²⋊₁₆C₂, C₄².₂₈C₂².₃C₂, C₂².₅₇C₂⁴.₂C₂, (C₂×C₄).₂₂₃(C₂×D₄), SmallGroup(128,396)

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

Derived series

C₁ — C₄² — C₄².₁₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄².C₂ — C₂².₅₇C₂⁴ — C₄².₁₀C₂³

Lower central

C₁ — C₂² — C₄² — C₄².₁₀C₂³

Upper central

C₁ — C₂² — C₄² — C₄².₁₀C₂³

Jennings

C₁ — C₂² — C₂² — C₄² — C₄².₁₀C₂³

Generators and relations for C₄².₁₀C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=a², d²=e²=a²b², ab=ba, cac^-1=dad^-1=a^-1, eae^-1=a^-1b², cbc^-1=ebe^-1=b^-1, dbd^-1=a²b^-1, dcd^-1=ac, ece^-1=bc, de=ed >

Subgroups: 208 in 92 conjugacy classes, 30 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₄⋊Q₈, C₄².C₂², C₄².₂C₂², C₄².₂₈C₂², C₄².₃₀C₂², C₈⋊Q₈, C₂².₅₇C₂⁴, C₄².₁₀C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, D₄.₈D₄, D₄.₉D₄, D₄.₁₀D₄, C₄².₁₀C₂³

Character table of C₄².₁₀C₂³

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F
size	1	1	1	1	8	4	4	4	8	8	8	8	8	16	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	trivial
ρ₂	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	2	2	-2	-2	0	0	0	0	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	-2	2	-2	0	2	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	-2	0	0	0	0	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	-2	2	-2	0	-2	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	-2	2	2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	-2	-2	2	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	2	0	-2	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₁₆	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	0	2	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₁₇	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	complex lifted from D₄.₉D₄
ρ₁₈	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	2i	0	0	-2i	0	complex lifted from D₄.₈D₄
ρ₁₉	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	complex lifted from D₄.₉D₄
ρ₂₀	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	-2i	0	0	2i	0	complex lifted from D₄.₈D₄

Smallest permutation representation of C₄².₁₀C₂³
►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 20 13 5)(2 17 14 6)(3 18 15 7)(4 19 16 8)(9 62 54 57)(10 63 55 58)(11 64 56 59)(12 61 53 60)(21 29 35 26)(22 30 36 27)(23 31 33 28)(24 32 34 25)(37 44 47 52)(38 41 48 49)(39 42 45 50)(40 43 46 51)

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

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

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

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

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

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

Matrix representation of C₄².₁₀C₂³ ►in GL₈(𝔽₁₇)

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

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

15	2	2	15	0	0	0	0
2	2	15	15	0	0	0	0
2	15	2	15	0	0	0	0
15	15	15	15	0	0	0	0
0	0	0	0	12	1	16	12
0	0	0	0	1	5	12	1
0	0	0	0	16	12	12	1
0	0	0	0	12	1	1	5

13	0	0	10	0	0	0	0
0	13	7	0	0	0	0	0
0	10	4	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	10	1	0	0
0	0	0	0	1	7	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

13	0	0	10	0	0	0	0
0	4	10	0	0	0	0	0
0	7	13	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	7	16	0	0
0	0	0	0	16	10	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[15,2,2,15,0,0,0,0,2,2,15,15,0,0,0,0,2,15,2,15,0,0,0,0,15,15,15,15,0,0,0,0,0,0,0,0,12,1,16,12,0,0,0,0,1,5,12,1,0,0,0,0,16,12,12,1,0,0,0,0,12,1,1,5],[13,0,0,7,0,0,0,0,0,13,10,0,0,0,0,0,0,7,4,0,0,0,0,0,10,0,0,4,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,1,7],[13,0,0,7,0,0,0,0,0,4,7,0,0,0,0,0,0,10,13,0,0,0,0,0,10,0,0,4,0,0,0,0,0,0,0,0,7,16,0,0,0,0,0,0,16,10,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,1,7] >;

C₄².₁₀C₂³ in GAP, Magma, Sage, TeX

C_4^2._{10}C_2^3

% in TeX

G:=Group("C4^2.10C2^3");

// GroupNames label

G:=SmallGroup(128,396);

// by ID

G=gap.SmallGroup(128,396);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,352,1123,570,521,136,3924,1411,998,242]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄².₁₀C₂³ in TeX

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

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

15	2	2	15	0	0	0	0
2	2	15	15	0	0	0	0
2	15	2	15	0	0	0	0
15	15	15	15	0	0	0	0
0	0	0	0	12	1	16	12
0	0	0	0	1	5	12	1
0	0	0	0	16	12	12	1
0	0	0	0	12	1	1	5

13	0	0	10	0	0	0	0
0	13	7	0	0	0	0	0
0	10	4	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	10	1	0	0
0	0	0	0	1	7	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

13	0	0	10	0	0	0	0
0	4	10	0	0	0	0	0
0	7	13	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	7	16	0	0
0	0	0	0	16	10	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

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

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

15	2	2	15	0	0	0	0
2	2	15	15	0	0	0	0
2	15	2	15	0	0	0	0
15	15	15	15	0	0	0	0
0	0	0	0	12	1	16	12
0	0	0	0	1	5	12	1
0	0	0	0	16	12	12	1
0	0	0	0	12	1	1	5

13	0	0	10	0	0	0	0
0	13	7	0	0	0	0	0
0	10	4	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	10	1	0	0
0	0	0	0	1	7	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

13	0	0	10	0	0	0	0
0	4	10	0	0	0	0	0
0	7	13	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	7	16	0	0
0	0	0	0	16	10	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

G = C42.10C23 order 128 = 27

10th non-split extension by C42 of C23 acting faithfully

G = C₄².₁₀C₂³ order 128 = 2⁷

10^th non-split extension by C₄² of C₂³ acting faithfully

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

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

15	2	2	15	0	0	0	0
2	2	15	15	0	0	0	0
2	15	2	15	0	0	0	0
15	15	15	15	0	0	0	0
0	0	0	0	12	1	16	12
0	0	0	0	1	5	12	1
0	0	0	0	16	12	12	1
0	0	0	0	12	1	1	5

13	0	0	10	0	0	0	0
0	13	7	0	0	0	0	0
0	10	4	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	10	1	0	0
0	0	0	0	1	7	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7

13	0	0	10	0	0	0	0
0	4	10	0	0	0	0	0
0	7	13	0	0	0	0	0
7	0	0	4	0	0	0	0
0	0	0	0	7	16	0	0
0	0	0	0	16	10	0	0
0	0	0	0	0	0	10	1
0	0	0	0	0	0	1	7