C4xC4:F5 - GroupNames

G = C₄×C₄⋊F₅ order 320 = 2⁶·5

Direct product of C₄ and C₄⋊F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄×C₄⋊F₅, C₄²⋊₇F₅, C₂₀⋊₂C₄², Dic₅⋊₅C₄², C₄⋊₁(C₄×F₅), (C₄×C₂₀)⋊₉C₄, C₂₀⋊₃(C₄⋊C₄), D₅.₁(C₄×D₄), D₅.₁(C₄×Q₈), (C₄×D₅).₈₈D₄, (C₄×D₅).₃₁Q₈, D₁₀.₇(C₂×Q₈), Dic₅⋊₅(C₄⋊C₄), (C₄×Dic₅)⋊₂₀C₄, D₁₀.₂₄(C₂×D₄), C₁₀.₆(C₂×C₄²), (D₅×C₄²).₂₇C₂, D₁₀.₂₂(C₄○D₄), D₁₀.₂₈(C₂²×C₄), C₁₀.₆(C₄²⋊C₂), C₂².₃₂(C₂²×F₅), D₁₀.₃Q₈.₁₀C₂, (C₂²×F₅).₁₃C₂², (C₂²×D₅).₂₆₄C₂³, C₂.₄(D₁₀.C₂³), C₅⋊₁(C₄×C₄⋊C₄), C₂.₈(C₂×C₄×F₅), C₂.₂(C₂×C₄⋊F₅), (C₂×C₄×F₅).₉C₂, C₁₀.₄(C₂×C₄⋊C₄), (C₂×C₄⋊F₅).₁₅C₂, (C₂×F₅).₁(C₂×C₄), (C₄×D₅).₆₅(C₂×C₄), (C₂×C₄).₁₆₂(C₂×F₅), (C₂×C₂₀).₁₇₁(C₂×C₄), (C₂×C₄×D₅).₃₉₂C₂², (C₂×C₁₀).₂₅(C₂²×C₄), (C₂×Dic₅).₁₇₃(C₂×C₄), SmallGroup(320,1025)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₄×C₄⋊F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — C₂×C₄⋊F₅ — C₄×C₄⋊F₅

Lower central

C₅ — C₁₀ — C₄×C₄⋊F₅

Upper central

C₁ — C₂×C₄ — C₄²

Generators and relations for C₄×C₄⋊F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c³ >

Subgroups: 666 in 194 conjugacy classes, 86 normal (32 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄², C₄², C₄⋊C₄, C₂²×C₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂.C₄², C₂×C₄², C₂×C₄⋊C₄, C₄×D₅, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₄×C₄⋊C₄, C₄×Dic₅, C₄×C₂₀, C₄×F₅, C₄⋊F₅, C₂×C₄×D₅, C₂²×F₅, D₁₀.₃Q₈, D₅×C₄², C₂×C₄×F₅, C₂×C₄⋊F₅, C₄×C₄⋊F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, F₅, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂×F₅, C₄×C₄⋊C₄, C₄×F₅, C₄⋊F₅, C₂²×F₅, C₂×C₄×F₅, C₂×C₄⋊F₅, D₁₀.C₂³, C₄×C₄⋊F₅

Smallest permutation representation of C₄×C₄⋊F₅
►On 80 points

Generators in S₈₀

(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)

(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 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)

(1 24 14 34)(2 21 13 32)(3 23 12 35)(4 25 11 33)(5 22 15 31)(6 28 17 40)(7 30 16 38)(8 27 20 36)(9 29 19 39)(10 26 18 37)(41 68 52 80)(42 70 51 78)(43 67 55 76)(44 69 54 79)(45 66 53 77)(46 63 57 75)(47 65 56 73)(48 62 60 71)(49 64 59 74)(50 61 58 72)

G:=sub<Sym(80)| (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,24,14,34)(2,21,13,32)(3,23,12,35)(4,25,11,33)(5,22,15,31)(6,28,17,40)(7,30,16,38)(8,27,20,36)(9,29,19,39)(10,26,18,37)(41,68,52,80)(42,70,51,78)(43,67,55,76)(44,69,54,79)(45,66,53,77)(46,63,57,75)(47,65,56,73)(48,62,60,71)(49,64,59,74)(50,61,58,72)>;

G:=Group( (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,24,14,34)(2,21,13,32)(3,23,12,35)(4,25,11,33)(5,22,15,31)(6,28,17,40)(7,30,16,38)(8,27,20,36)(9,29,19,39)(10,26,18,37)(41,68,52,80)(42,70,51,78)(43,67,55,76)(44,69,54,79)(45,66,53,77)(46,63,57,75)(47,65,56,73)(48,62,60,71)(49,64,59,74)(50,61,58,72) );

G=PermutationGroup([[(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(1,59,9,54),(2,60,10,55),(3,56,6,51),(4,57,7,52),(5,58,8,53),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,76,26,71),(22,77,27,72),(23,78,28,73),(24,79,29,74),(25,80,30,75),(31,66,36,61),(32,67,37,62),(33,68,38,63),(34,69,39,64),(35,70,40,65)], [(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),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,24,14,34),(2,21,13,32),(3,23,12,35),(4,25,11,33),(5,22,15,31),(6,28,17,40),(7,30,16,38),(8,27,20,36),(9,29,19,39),(10,26,18,37),(41,68,52,80),(42,70,51,78),(43,67,55,76),(44,69,54,79),(45,66,53,77),(46,63,57,75),(47,65,56,73),(48,62,60,71),(49,64,59,74),(50,61,58,72)]])

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	···	4AF	5	10A	10B	10C	20A	···	20L
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	4	···	4	5	10	10	10	20	···	20
size	1	1	1	1	5	5	5	5	1	1	1	1	2	2	2	2	5	5	5	5	10	···	10	4	4	4	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4	4	4	4
type	+	+	+	+	+				+	-		+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	C₄○D₄	F₅	C₂×F₅	C₄×F₅	C₄⋊F₅	D₁₀.C₂³
kernel	C₄×C₄⋊F₅	D₁₀.₃Q₈	D₅×C₄²	C₂×C₄×F₅	C₂×C₄⋊F₅	C₄×Dic₅	C₄×C₂₀	C₄⋊F₅	C₄×D₅	C₄×D₅	D₁₀	C₄²	C₂×C₄	C₄	C₄	C₂
# reps	1	2	1	2	2	6	2	16	2	2	4	1	3	4	4	4

Matrix representation of C₄×C₄⋊F₅ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

0	40	0	0	0	0
1	0	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	1	0	0
0	0	40	0	1	0
0	0	40	0	0	1
0	0	40	0	0	0

9	0	0	0	0	0
0	32	0	0	0	0
0	0	0	0	32	0
0	0	32	0	0	0
0	0	0	0	0	32
0	0	0	32	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C₄×C₄⋊F₅ in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes F_5

% in TeX

G:=Group("C4xC4:F5");

// GroupNames label

G:=SmallGroup(320,1025);

// by ID

G=gap.SmallGroup(320,1025);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,100,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;

// generators/relations

G = C4×C4⋊F5 order 320 = 26·5

Direct product of C4 and C4⋊F5

G = C₄×C₄⋊F₅ order 320 = 2⁶·5

Direct product of C₄ and C₄⋊F₅