Copied to
clipboard

## G = C2×C29⋊C4order 232 = 23·29

### Direct product of C2 and C29⋊C4

Aliases: C2×C29⋊C4, C58⋊C4, D29⋊C4, D58.C2, D29.C22, C29⋊(C2×C4), SmallGroup(232,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — C2×C29⋊C4
 Chief series C1 — C29 — D29 — C29⋊C4 — C2×C29⋊C4
 Lower central C29 — C2×C29⋊C4
 Upper central C1 — C2

Generators and relations for C2×C29⋊C4
G = < a,b,c | a2=b29=c4=1, ab=ba, ac=ca, cbc-1=b17 >

Character table of C2×C29⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 29A 29B 29C 29D 29E 29F 29G 58A 58B 58C 58D 58E 58F 58G size 1 1 29 29 29 29 29 29 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 -1 -1 1 i -i -i i 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 -i i i -i 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 4 -4 0 0 0 0 0 0 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 -ζ2921-ζ2920-ζ299-ζ298 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2927-ζ2924-ζ295-ζ292 -ζ2926-ζ2922-ζ297-ζ293 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2923-ζ2915-ζ2914-ζ296 orthogonal faithful ρ10 4 4 0 0 0 0 0 0 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 orthogonal lifted from C29⋊C4 ρ11 4 4 0 0 0 0 0 0 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 orthogonal lifted from C29⋊C4 ρ12 4 4 0 0 0 0 0 0 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 orthogonal lifted from C29⋊C4 ρ13 4 -4 0 0 0 0 0 0 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 -ζ2927-ζ2924-ζ295-ζ292 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2926-ζ2922-ζ297-ζ293 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2921-ζ2920-ζ299-ζ298 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2918-ζ2916-ζ2913-ζ2911 orthogonal faithful ρ14 4 4 0 0 0 0 0 0 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 orthogonal lifted from C29⋊C4 ρ15 4 4 0 0 0 0 0 0 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 orthogonal lifted from C29⋊C4 ρ16 4 4 0 0 0 0 0 0 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 orthogonal lifted from C29⋊C4 ρ17 4 4 0 0 0 0 0 0 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 orthogonal lifted from C29⋊C4 ρ18 4 -4 0 0 0 0 0 0 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2921-ζ2920-ζ299-ζ298 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2927-ζ2924-ζ295-ζ292 -ζ2926-ζ2922-ζ297-ζ293 -ζ2925-ζ2919-ζ2910-ζ294 orthogonal faithful ρ19 4 -4 0 0 0 0 0 0 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2921-ζ2920-ζ299-ζ298 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2927-ζ2924-ζ295-ζ292 -ζ2926-ζ2922-ζ297-ζ293 orthogonal faithful ρ20 4 -4 0 0 0 0 0 0 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 -ζ2926-ζ2922-ζ297-ζ293 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2921-ζ2920-ζ299-ζ298 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2927-ζ2924-ζ295-ζ292 orthogonal faithful ρ21 4 -4 0 0 0 0 0 0 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 -ζ2928-ζ2917-ζ2912-ζ29 -ζ2927-ζ2924-ζ295-ζ292 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2926-ζ2922-ζ297-ζ293 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2921-ζ2920-ζ299-ζ298 orthogonal faithful ρ22 4 -4 0 0 0 0 0 0 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 -ζ2918-ζ2916-ζ2913-ζ2911 -ζ2926-ζ2922-ζ297-ζ293 -ζ2927-ζ2924-ζ295-ζ292 -ζ2925-ζ2919-ζ2910-ζ294 -ζ2923-ζ2915-ζ2914-ζ296 -ζ2921-ζ2920-ζ299-ζ298 -ζ2928-ζ2917-ζ2912-ζ29 orthogonal faithful

Smallest permutation representation of C2×C29⋊C4
On 58 points
Generators in S58
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 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)
(2 13 29 18)(3 25 28 6)(4 8 27 23)(5 20 26 11)(7 15 24 16)(9 10 22 21)(12 17 19 14)(31 42 58 47)(32 54 57 35)(33 37 56 52)(34 49 55 40)(36 44 53 45)(38 39 51 50)(41 46 48 43)

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

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

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

C2×C29⋊C4 is a maximal subgroup of   C116⋊C4  D29.D4
C2×C29⋊C4 is a maximal quotient of   D29⋊C8  C116.C4  C116⋊C4  C29⋊M4(2)  D29.D4

Matrix representation of C2×C29⋊C4 in GL4(𝔽233) generated by

 232 0 0 0 0 232 0 0 0 0 232 0 0 0 0 232
,
 209 24 30 232 210 24 30 232 209 25 30 232 209 24 31 232
,
 51 61 23 120 203 209 24 30 230 92 229 210 6 83 163 210
G:=sub<GL(4,GF(233))| [232,0,0,0,0,232,0,0,0,0,232,0,0,0,0,232],[209,210,209,209,24,24,25,24,30,30,30,31,232,232,232,232],[51,203,230,6,61,209,92,83,23,24,229,163,120,30,210,210] >;

C2×C29⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{29}\rtimes C_4
% in TeX

G:=Group("C2xC29:C4");
// GroupNames label

G:=SmallGroup(232,12);
// by ID

G=gap.SmallGroup(232,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-29,16,1539,907]);
// Polycyclic

G:=Group<a,b,c|a^2=b^29=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations

Export

׿
×
𝔽