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G = C4○D4×D13order 416 = 25·13

Direct product of C4○D4 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C4○D4×D13
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C2×C4×D13 — C4○D4×D13
 Lower central C13 — C26 — C4○D4×D13
 Upper central C1 — C4 — C4○D4

Generators and relations for C4○D4×D13
G = < a,b,c,d,e | a4=c2=d13=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1040 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2 [×8], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C13, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], D13 [×2], D13 [×3], C26, C26 [×3], C2×C4○D4, Dic13, Dic13 [×3], C52, C52 [×3], D26, D26 [×3], D26 [×6], C2×C26 [×3], Dic26 [×3], C4×D13, C4×D13 [×9], D52 [×3], C2×Dic13 [×3], C13⋊D4 [×6], C2×C52 [×3], D4×C13 [×3], Q8×C13, C22×D13 [×3], C2×C4×D13 [×3], D525C2 [×3], D4×D13 [×3], D42D13 [×3], Q8×D13, D52⋊C2, C13×C4○D4, C4○D4×D13
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, D13, C2×C4○D4, D26 [×7], C22×D13 [×7], C23×D13, C4○D4×D13

Smallest permutation representation of C4○D4×D13
On 104 points
Generators in S104
(1 40 24 32)(2 41 25 33)(3 42 26 34)(4 43 14 35)(5 44 15 36)(6 45 16 37)(7 46 17 38)(8 47 18 39)(9 48 19 27)(10 49 20 28)(11 50 21 29)(12 51 22 30)(13 52 23 31)(53 100 75 84)(54 101 76 85)(55 102 77 86)(56 103 78 87)(57 104 66 88)(58 92 67 89)(59 93 68 90)(60 94 69 91)(61 95 70 79)(62 96 71 80)(63 97 72 81)(64 98 73 82)(65 99 74 83)
(1 56 24 78)(2 57 25 66)(3 58 26 67)(4 59 14 68)(5 60 15 69)(6 61 16 70)(7 62 17 71)(8 63 18 72)(9 64 19 73)(10 65 20 74)(11 53 21 75)(12 54 22 76)(13 55 23 77)(27 82 48 98)(28 83 49 99)(29 84 50 100)(30 85 51 101)(31 86 52 102)(32 87 40 103)(33 88 41 104)(34 89 42 92)(35 90 43 93)(36 91 44 94)(37 79 45 95)(38 80 46 96)(39 81 47 97)
(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 92)(90 93)(91 94)
(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 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 20)(15 19)(16 18)(21 26)(22 25)(23 24)(27 36)(28 35)(29 34)(30 33)(31 32)(37 39)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 58)(54 57)(55 56)(59 65)(60 64)(61 63)(66 76)(67 75)(68 74)(69 73)(70 72)(77 78)(79 81)(82 91)(83 90)(84 89)(85 88)(86 87)(92 100)(93 99)(94 98)(95 97)(101 104)(102 103)

G:=sub<Sym(104)| (1,40,24,32)(2,41,25,33)(3,42,26,34)(4,43,14,35)(5,44,15,36)(6,45,16,37)(7,46,17,38)(8,47,18,39)(9,48,19,27)(10,49,20,28)(11,50,21,29)(12,51,22,30)(13,52,23,31)(53,100,75,84)(54,101,76,85)(55,102,77,86)(56,103,78,87)(57,104,66,88)(58,92,67,89)(59,93,68,90)(60,94,69,91)(61,95,70,79)(62,96,71,80)(63,97,72,81)(64,98,73,82)(65,99,74,83), (1,56,24,78)(2,57,25,66)(3,58,26,67)(4,59,14,68)(5,60,15,69)(6,61,16,70)(7,62,17,71)(8,63,18,72)(9,64,19,73)(10,65,20,74)(11,53,21,75)(12,54,22,76)(13,55,23,77)(27,82,48,98)(28,83,49,99)(29,84,50,100)(30,85,51,101)(31,86,52,102)(32,87,40,103)(33,88,41,104)(34,89,42,92)(35,90,43,93)(36,91,44,94)(37,79,45,95)(38,80,46,96)(39,81,47,97), (53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,92)(90,93)(91,94), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,36)(28,35)(29,34)(30,33)(31,32)(37,39)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,81)(82,91)(83,90)(84,89)(85,88)(86,87)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103)>;

G:=Group( (1,40,24,32)(2,41,25,33)(3,42,26,34)(4,43,14,35)(5,44,15,36)(6,45,16,37)(7,46,17,38)(8,47,18,39)(9,48,19,27)(10,49,20,28)(11,50,21,29)(12,51,22,30)(13,52,23,31)(53,100,75,84)(54,101,76,85)(55,102,77,86)(56,103,78,87)(57,104,66,88)(58,92,67,89)(59,93,68,90)(60,94,69,91)(61,95,70,79)(62,96,71,80)(63,97,72,81)(64,98,73,82)(65,99,74,83), (1,56,24,78)(2,57,25,66)(3,58,26,67)(4,59,14,68)(5,60,15,69)(6,61,16,70)(7,62,17,71)(8,63,18,72)(9,64,19,73)(10,65,20,74)(11,53,21,75)(12,54,22,76)(13,55,23,77)(27,82,48,98)(28,83,49,99)(29,84,50,100)(30,85,51,101)(31,86,52,102)(32,87,40,103)(33,88,41,104)(34,89,42,92)(35,90,43,93)(36,91,44,94)(37,79,45,95)(38,80,46,96)(39,81,47,97), (53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,92)(90,93)(91,94), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,36)(28,35)(29,34)(30,33)(31,32)(37,39)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,81)(82,91)(83,90)(84,89)(85,88)(86,87)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103) );

G=PermutationGroup([(1,40,24,32),(2,41,25,33),(3,42,26,34),(4,43,14,35),(5,44,15,36),(6,45,16,37),(7,46,17,38),(8,47,18,39),(9,48,19,27),(10,49,20,28),(11,50,21,29),(12,51,22,30),(13,52,23,31),(53,100,75,84),(54,101,76,85),(55,102,77,86),(56,103,78,87),(57,104,66,88),(58,92,67,89),(59,93,68,90),(60,94,69,91),(61,95,70,79),(62,96,71,80),(63,97,72,81),(64,98,73,82),(65,99,74,83)], [(1,56,24,78),(2,57,25,66),(3,58,26,67),(4,59,14,68),(5,60,15,69),(6,61,16,70),(7,62,17,71),(8,63,18,72),(9,64,19,73),(10,65,20,74),(11,53,21,75),(12,54,22,76),(13,55,23,77),(27,82,48,98),(28,83,49,99),(29,84,50,100),(30,85,51,101),(31,86,52,102),(32,87,40,103),(33,88,41,104),(34,89,42,92),(35,90,43,93),(36,91,44,94),(37,79,45,95),(38,80,46,96),(39,81,47,97)], [(53,75),(54,76),(55,77),(56,78),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(65,74),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104),(89,92),(90,93),(91,94)], [(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),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,20),(15,19),(16,18),(21,26),(22,25),(23,24),(27,36),(28,35),(29,34),(30,33),(31,32),(37,39),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,58),(54,57),(55,56),(59,65),(60,64),(61,63),(66,76),(67,75),(68,74),(69,73),(70,72),(77,78),(79,81),(82,91),(83,90),(84,89),(85,88),(86,87),(92,100),(93,99),(94,98),(95,97),(101,104),(102,103)])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26F 26G ··· 26X 52A ··· 52L 52M ··· 52AD order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 2 2 13 13 26 26 26 1 1 2 2 2 13 13 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4○D4 D13 D26 D26 D26 C4○D4×D13 kernel C4○D4×D13 C2×C4×D13 D52⋊5C2 D4×D13 D4⋊2D13 Q8×D13 D52⋊C2 C13×C4○D4 D13 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 3 3 1 1 1 4 6 18 18 6 12

Matrix representation of C4○D4×D13 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 30 0 0 0 0 30
,
 52 0 0 0 0 52 0 0 0 0 52 30 0 0 7 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 46 52
,
 32 1 0 0 16 32 0 0 0 0 1 0 0 0 0 1
,
 13 39 0 0 12 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,30,0,0,0,0,30],[52,0,0,0,0,52,0,0,0,0,52,7,0,0,30,1],[1,0,0,0,0,1,0,0,0,0,1,46,0,0,0,52],[32,16,0,0,1,32,0,0,0,0,1,0,0,0,0,1],[13,12,0,0,39,40,0,0,0,0,1,0,0,0,0,1] >;

C4○D4×D13 in GAP, Magma, Sage, TeX

C_4\circ D_4\times D_{13}
% in TeX

G:=Group("C4oD4xD13");
// GroupNames label

G:=SmallGroup(416,222);
// by ID

G=gap.SmallGroup(416,222);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,297,13829]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^13=e^2=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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