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## G = C22×C7⋊D4order 224 = 25·7

### Direct product of C22 and C7⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C22×C7⋊D4
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — C22×C7⋊D4
 Lower central C7 — C14 — C22×C7⋊D4
 Upper central C1 — C23 — C24

Generators and relations for C22×C7⋊D4
G = < a,b,c,d,e | a2=b2=c7=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 862 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22×C4, C2×D4, C24, C24, Dic7, D14, D14, C2×C14, C2×C14, C22×D4, C2×Dic7, C7⋊D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C22×Dic7, C2×C7⋊D4, C23×D7, C23×C14, C22×C7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C7⋊D4, C22×D7, C2×C7⋊D4, C23×D7, C22×C7⋊D4

Smallest permutation representation of C22×C7⋊D4
On 112 points
Generators in S112
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(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)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(1 71 8 78)(2 77 9 84)(3 76 10 83)(4 75 11 82)(5 74 12 81)(6 73 13 80)(7 72 14 79)(15 64 22 57)(16 70 23 63)(17 69 24 62)(18 68 25 61)(19 67 26 60)(20 66 27 59)(21 65 28 58)(29 99 36 106)(30 105 37 112)(31 104 38 111)(32 103 39 110)(33 102 40 109)(34 101 41 108)(35 100 42 107)(43 92 50 85)(44 98 51 91)(45 97 52 90)(46 96 53 89)(47 95 54 88)(48 94 55 87)(49 93 56 86)
(1 85)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 92)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 106)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 99)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 57)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 64)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 78)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 71)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)

G:=sub<Sym(112)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,71,8,78)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,64,22,57)(16,70,23,63)(17,69,24,62)(18,68,25,61)(19,67,26,60)(20,66,27,59)(21,65,28,58)(29,99,36,106)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,92,50,85)(44,98,51,91)(45,97,52,90)(46,96,53,89)(47,95,54,88)(48,94,55,87)(49,93,56,86), (1,85)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,92)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,106)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,99)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,57)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,64)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,78)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,71)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)>;

G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,71,8,78)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,64,22,57)(16,70,23,63)(17,69,24,62)(18,68,25,61)(19,67,26,60)(20,66,27,59)(21,65,28,58)(29,99,36,106)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,92,50,85)(44,98,51,91)(45,97,52,90)(46,96,53,89)(47,95,54,88)(48,94,55,87)(49,93,56,86), (1,85)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,92)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,106)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,99)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,57)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,64)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,78)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,71)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72) );

G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(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),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,71,8,78),(2,77,9,84),(3,76,10,83),(4,75,11,82),(5,74,12,81),(6,73,13,80),(7,72,14,79),(15,64,22,57),(16,70,23,63),(17,69,24,62),(18,68,25,61),(19,67,26,60),(20,66,27,59),(21,65,28,58),(29,99,36,106),(30,105,37,112),(31,104,38,111),(32,103,39,110),(33,102,40,109),(34,101,41,108),(35,100,42,107),(43,92,50,85),(44,98,51,91),(45,97,52,90),(46,96,53,89),(47,95,54,88),(48,94,55,87),(49,93,56,86)], [(1,85),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,92),(9,98),(10,97),(11,96),(12,95),(13,94),(14,93),(15,106),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,99),(23,105),(24,104),(25,103),(26,102),(27,101),(28,100),(29,57),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,64),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,78),(44,84),(45,83),(46,82),(47,81),(48,80),(49,79),(50,71),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72)]])

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 7A 7B 7C 14A ··· 14AS order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 4 7 7 7 14 ··· 14 size 1 1 ··· 1 2 2 2 2 14 14 14 14 14 14 14 14 2 2 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D7 D14 C7⋊D4 kernel C22×C7⋊D4 C22×Dic7 C2×C7⋊D4 C23×D7 C23×C14 C2×C14 C24 C23 C22 # reps 1 1 12 1 1 4 3 21 24

Matrix representation of C22×C7⋊D4 in GL5(𝔽29)

 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 25 28 0 0 0 2 22 0 0 0 0 0 4 28 0 0 0 5 28
,
 28 0 0 0 0 0 14 11 0 0 0 6 15 0 0 0 0 0 7 24 0 0 0 10 22
,
 1 0 0 0 0 0 14 11 0 0 0 6 15 0 0 0 0 0 25 1 0 0 0 14 4

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,25,2,0,0,0,28,22,0,0,0,0,0,4,5,0,0,0,28,28],[28,0,0,0,0,0,14,6,0,0,0,11,15,0,0,0,0,0,7,10,0,0,0,24,22],[1,0,0,0,0,0,14,6,0,0,0,11,15,0,0,0,0,0,25,14,0,0,0,1,4] >;

C22×C7⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes D_4
% in TeX

G:=Group("C2^2xC7:D4");
// GroupNames label

G:=SmallGroup(224,188);
// by ID

G=gap.SmallGroup(224,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,579,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^7=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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