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## G = D40.6C4order 320 = 26·5

### 4th non-split extension by D40 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D40.6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C2×D40 — D40.6C4
 Lower central C5 — C10 — C20 — C40 — D40.6C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for D40.6C4
G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a11, cbc-1=a15b >

Subgroups: 414 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, C2×D4, C20, D10, C2×C10, C8.C4, M5(2), C2×D8, C40, C40, D20, C2×C20, C22×D5, M5(2)⋊C2, C52C16, D40, D40, C2×C40, C5×M4(2), C2×D20, C20.4C8, C5×C8.C4, C2×D40, D40.6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, M5(2)⋊C2, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D40.6C4

Smallest permutation representation of D40.6C4
On 80 points
Generators in S80
(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 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(72 80)(73 79)(74 78)(75 77)
(1 69 11 59 21 49 31 79)(2 80 12 70 22 60 32 50)(3 51 13 41 23 71 33 61)(4 62 14 52 24 42 34 72)(5 73 15 63 25 53 35 43)(6 44 16 74 26 64 36 54)(7 55 17 45 27 75 37 65)(8 66 18 56 28 46 38 76)(9 77 19 67 29 57 39 47)(10 48 20 78 30 68 40 58)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C 10D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I ··· 40P order 1 2 2 2 2 4 4 5 5 8 8 8 8 8 10 10 10 10 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 ··· 40 size 1 1 2 40 40 2 2 2 2 2 2 4 8 8 2 2 4 4 20 20 20 20 2 2 2 2 4 4 4 ··· 4 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 C4×D5 D20 C5⋊D4 M5(2)⋊C2 D4⋊D5 Q8⋊D5 D40.6C4 kernel D40.6C4 C20.4C8 C5×C8.C4 C2×D40 D40 C40 C2×C20 C8.C4 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of D40.6C4 in GL4(𝔽241) generated by

 29 185 0 0 76 194 0 0 7 182 56 27 35 172 214 228
,
 227 27 0 0 207 14 0 0 13 224 190 190 94 109 240 51
,
 0 0 240 1 126 196 239 51 19 61 45 0 219 58 45 0
G:=sub<GL(4,GF(241))| [29,76,7,35,185,194,182,172,0,0,56,214,0,0,27,228],[227,207,13,94,27,14,224,109,0,0,190,240,0,0,190,51],[0,126,19,219,0,196,61,58,240,239,45,45,1,51,0,0] >;

D40.6C4 in GAP, Magma, Sage, TeX

D_{40}._6C_4
% in TeX

G:=Group("D40.6C4");
// GroupNames label

G:=SmallGroup(320,53);
// by ID

G=gap.SmallGroup(320,53);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,184,675,794,192,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

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