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## G = C7×C24⋊C22order 448 = 26·7

### Direct product of C7 and C24⋊C22

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C24⋊C22
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — Q8×C14 — C7×C4.4D4 — C7×C24⋊C22
 Lower central C1 — C22 — C7×C24⋊C22
 Upper central C1 — C2×C14 — C7×C24⋊C22

Generators and relations for C7×C24⋊C22
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 506 in 260 conjugacy classes, 142 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, C24, C28, C2×C14, C2×C14, C22≀C2, C4.4D4, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C24⋊C22, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C23×C14, C7×C22≀C2, C7×C4.4D4, C7×C24⋊C22
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2+ 1+4, C22×C14, C24⋊C22, C23×C14, C7×2+ 1+4, C7×C24⋊C22

Smallest permutation representation of C7×C24⋊C22
On 112 points
Generators in S112
(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 94)(2 95)(3 96)(4 97)(5 98)(6 92)(7 93)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 64)(15 59)(16 60)(17 61)(18 62)(19 63)(20 57)(21 58)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 82)(37 83)(38 84)(39 78)(40 79)(41 80)(42 81)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(71 109)(72 110)(73 111)(74 112)(75 106)(76 107)(77 108)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 64)(7 65)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 92)(15 87)(16 88)(17 89)(18 90)(19 91)(20 85)(21 86)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 54)(37 55)(38 56)(39 50)(40 51)(41 52)(42 53)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(99 109)(100 110)(101 111)(102 112)(103 106)(104 107)(105 108)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(50 63)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)
(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)

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

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

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

133 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4I 7A ··· 7F 14A ··· 14R 14S ··· 14BB 28A ··· 28BB order 1 2 2 2 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 ··· 4 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

133 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C7 C14 C14 2+ 1+4 C7×2+ 1+4 kernel C7×C24⋊C22 C7×C22≀C2 C7×C4.4D4 C24⋊C22 C22≀C2 C4.4D4 C14 C2 # reps 1 6 9 6 36 54 3 18

Matrix representation of C7×C24⋊C22 in GL8(𝔽29)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0
,
 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28

G:=sub<GL(8,GF(29))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28] >;

C7×C24⋊C22 in GAP, Magma, Sage, TeX

C_7\times C_2^4\rtimes C_2^2
% in TeX

G:=Group("C7xC2^4:C2^2");
// GroupNames label

G:=SmallGroup(448,1344);
// by ID

G=gap.SmallGroup(448,1344);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,3579,9635,1690]);
// Polycyclic

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

׿
×
𝔽