C2^2xD4xD7 - GroupNames

G = C₂²×D₄×D₇ order 448 = 2⁶·7

Direct product of C₂², D₄ and D₇

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

Aliases: C₂²×D₄×D₇, C₂₈⋊C₂⁴, D₁₄⋊₂C₂⁴, D₂₈⋊₈C₂³, C₂⁴⋊₁₀D₁₄, C₁₄.₅C₂⁵, Dic₇⋊₁C₂⁴, (C₂×C₁₄)⋊C₂⁴, C₇⋊₂(D₄×C₂³), C₄⋊₁(C₂³×D₇), (C₂×C₂₈)⋊₄C₂³, (D₇×C₂⁴)⋊₅C₂, (C₄×D₇)⋊₄C₂³, (C₇×D₄)⋊₆C₂³, C₁₄⋊₂(C₂²×D₄), C₇⋊D₄⋊₁C₂³, C₂.₆(D₇×C₂⁴), (C₂²×C₄)⋊₃₉D₁₄, C₂³⋊₅(C₂²×D₇), C₂²⋊₁(C₂³×D₇), (D₄×C₁₄)⋊₄₉C₂², (C₂²×D₂₈)⋊₂₂C₂, (C₂×D₂₈)⋊₆₀C₂², (C₂²×C₁₄)⋊₆C₂³, (C₂²×D₇)⋊₈C₂³, (C₂²×C₂₈)⋊₂₅C₂², (C₂³×C₁₄)⋊₁₅C₂², (C₂×Dic₇)⋊₁₀C₂³, (C₂³×D₇)⋊₂₃C₂², (C₂²×Dic₇)⋊₅₁C₂², (D₄×C₂×C₁₄)⋊₉C₂, (D₇×C₂²×C₄)⋊₈C₂, (C₂×C₁₄)⋊₁₄(C₂×D₄), (C₂×C₄×D₇)⋊₅₈C₂², (C₂×C₄)⋊₈(C₂²×D₇), (C₂×C₇⋊D₄)⋊₅₀C₂², (C₂²×C₇⋊D₄)⋊₁₉C₂, SmallGroup(448,1369)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂²×D₄×D₇

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — D₇×C₂⁴ — C₂²×D₄×D₇

Lower central

C₇ — C₁₄ — C₂²×D₄×D₇

Upper central

C₁ — C₂³ — C₂²×D₄

Generators and relations for C₂²×D₄×D₇
G = < a,b,c,d,e,f | a²=b²=c⁴=d²=e⁷=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 6548 in 1362 conjugacy classes, 503 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₂³, D₇, D₇, C₁₄, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, D₄×C₂³, C₂×C₄×D₇, C₂×D₂₈, D₄×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, C₂³×D₇, C₂³×D₇, C₂³×D₇, C₂³×C₁₄, D₇×C₂²×C₄, C₂²×D₂₈, C₂×D₄×D₇, C₂²×C₇⋊D₄, D₄×C₂×C₁₄, D₇×C₂⁴, C₂²×D₄×D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²×D₄, C₂⁵, C₂²×D₇, D₄×C₂³, D₄×D₇, C₂³×D₇, C₂×D₄×D₇, D₇×C₂⁴, C₂²×D₄×D₇

Smallest permutation representation of C₂²×D₄×D₇
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

100 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	···	2W	2X	···	2AE	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	14A	···	14U	14V	···	14AS	28A	···	28L
order	1	2	···	2	2	···	2	2	···	2	2	···	2	4	4	4	4	4	4	4	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	···	1	2	···	2	7	···	7	14	···	14	2	2	2	2	14	14	14	14	2	2	2	2	···	2	4	···	4	4	···	4

100 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

D₄×D₇

kernel

C₂²×D₄×D₇

D₇×C₂²×C₄

C₂²×D₂₈

C₂×D₄×D₇

C₂²×C₇⋊D₄

D₄×C₂×C₁₄

D₇×C₂⁴

C₂²×D₇

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₂²

# reps

Matrix representation of C₂²×D₄×D₇ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	28	0
0	0	0	0	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	28	13
0	0	0	0	11	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	28	0
0	0	0	0	11	1

11	1	0	0	0	0
13	25	0	0	0	0
0	0	0	1	0	0
0	0	28	7	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	1	0	0	0	0
14	25	0	0	0	0
0	0	0	28	0	0
0	0	28	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,13,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,11,0,0,0,0,0,1],[11,13,0,0,0,0,1,25,0,0,0,0,0,0,0,28,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,14,0,0,0,0,1,25,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂²×D₄×D₇ in GAP, Magma, Sage, TeX

C_2^2\times D_4\times D_7

% in TeX

G:=Group("C2^2xD4xD7");

// GroupNames label

G:=SmallGroup(448,1369);

// by ID

G=gap.SmallGroup(448,1369);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,235,18822]);

// Polycyclic

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

// generators/relations

G = C22×D4×D7 order 448 = 26·7

Direct product of C22, D4 and D7

G = C₂²×D₄×D₇ order 448 = 2⁶·7

Direct product of C₂², D₄ and D₇