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G = C2×C22⋊C16order 128 = 27

Direct product of C2 and C22⋊C16

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C22⋊C16, C232C16, C24.4C8, C22.9M5(2), C4(C22⋊C16), C8(C22⋊C16), (C22×C16)⋊1C2, C222(C2×C16), C8.130(C2×D4), (C2×C8).402D4, (C23×C8).9C2, (C2×C16)⋊14C22, C2.1(C22×C16), (C23×C4).31C4, C23.35(C2×C8), (C22×C4).11C8, (C22×C8).29C4, C2.3(C2×M5(2)), C8.62(C22⋊C4), C4.37(C22⋊C8), (C2×C8).623C23, C4.61(C2×M4(2)), (C2×C4).92M4(2), C22.26(C22×C8), C22.43(C22⋊C8), (C22×C8).592C22, (C2×C4).64(C2×C8), C2.3(C2×C22⋊C8), (C2×C8)(C22⋊C16), (C2×C4)(C22⋊C16), (C2×C8).214(C2×C4), C4.112(C2×C22⋊C4), (C2×C4).608(C22×C4), (C22×C4).445(C2×C4), (C2×C4).402(C22⋊C4), SmallGroup(128,843)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C22⋊C16
C1C2C4C8C2×C8C22×C8C23×C8 — C2×C22⋊C16
C1C2 — C2×C22⋊C16
C1C22×C8 — C2×C22⋊C16
C1C2C2C2C2C4C4C2×C8 — C2×C22⋊C16

Generators and relations for C2×C22⋊C16
 G = < a,b,c,d | a2=b2=c2=d16=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 196 in 136 conjugacy classes, 76 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×6], C23 [×4], C16 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×10], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C2×C16 [×4], C2×C16 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C23×C4, C22⋊C16 [×4], C22×C16 [×2], C23×C8, C2×C22⋊C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C16 [×4], C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C16 [×6], M5(2) [×2], C2×C22⋊C4, C22×C8, C2×M4(2), C22⋊C16 [×4], C2×C22⋊C8, C22×C16, C2×M5(2), C2×C22⋊C16

Smallest permutation representation of C2×C22⋊C16
On 64 points
Generators in S64
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)
(1 9)(2 21)(3 11)(4 23)(5 13)(6 25)(7 15)(8 27)(10 29)(12 31)(14 17)(16 19)(18 26)(20 28)(22 30)(24 32)(33 50)(34 42)(35 52)(36 44)(37 54)(38 46)(39 56)(40 48)(41 58)(43 60)(45 62)(47 64)(49 57)(51 59)(53 61)(55 63)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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)

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

G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (1,9)(2,21)(3,11)(4,23)(5,13)(6,25)(7,15)(8,27)(10,29)(12,31)(14,17)(16,19)(18,26)(20,28)(22,30)(24,32)(33,50)(34,42)(35,52)(36,44)(37,54)(38,46)(39,56)(40,48)(41,58)(43,60)(45,62)(47,64)(49,57)(51,59)(53,61)(55,63), (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57), (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) );

G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55)], [(1,9),(2,21),(3,11),(4,23),(5,13),(6,25),(7,15),(8,27),(10,29),(12,31),(14,17),(16,19),(18,26),(20,28),(22,30),(24,32),(33,50),(34,42),(35,52),(36,44),(37,54),(38,46),(39,56),(40,48),(41,58),(43,60),(45,62),(47,64),(49,57),(51,59),(53,61),(55,63)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57)], [(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)])

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L8A···8P8Q···8X16A···16AF
order12···222224···444448···88···816···16
size11···122221···122221···12···22···2

80 irreducible representations

dim111111111222
type+++++
imageC1C2C2C2C4C4C8C8C16D4M4(2)M5(2)
kernelC2×C22⋊C16C22⋊C16C22×C16C23×C8C22×C8C23×C4C22×C4C24C23C2×C8C2×C4C22
# reps14216212432448

Matrix representation of C2×C22⋊C16 in GL4(𝔽17) generated by

1000
01600
00160
00016
,
1000
01600
00165
0001
,
1000
0100
00160
00016
,
12000
01600
0070
001310
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,5,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[12,0,0,0,0,16,0,0,0,0,7,13,0,0,0,10] >;

C2×C22⋊C16 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes C_{16}
% in TeX

G:=Group("C2xC2^2:C16");
// GroupNames label

G:=SmallGroup(128,843);
// by ID

G=gap.SmallGroup(128,843);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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