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G = D4×C22×C4order 128 = 27

Direct product of C22×C4 and D4

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

Aliases: D4×C22×C4, C4220C23, C22.8C25, C25.94C22, C23.103C24, C24.602C23, (C24×C4)⋊5C2, C41(C23×C4), C4⋊C423C23, C2416(C2×C4), C2.4(C24×C4), C2.3(D4×C23), C221(C23×C4), C236(C22×C4), C22⋊C421C23, (C2×C4).155C24, (C2×C42)⋊87C22, (C22×C42)⋊19C2, (C23×C4)⋊57C22, (C22×C4)⋊22C23, (D4×C23).22C2, C23.888(C2×D4), (C2×D4).494C23, C23.375(C4○D4), C22.156(C22×D4), (C22×D4).611C22, C4(C2×C4×D4), (C2×C4)2(C4×D4), C4⋊C4(C23×C4), C4⋊C43(C22×C4), C42(C22×C4⋊C4), (C22×C4)(C4×D4), (C2×C4)(D4×C23), (C2×D4)(C23×C4), (C2×C4)⋊9(C22×C4), C22⋊C4(C23×C4), (C22×C4⋊C4)⋊49C2, (C22×C4)⋊47(C2×C4), C22⋊C43(C22×C4), C42(C22×C22⋊C4), C2.2(C22×C4○D4), (C2×C4⋊C4)⋊144C22, (C23×C4)(C22×D4), (C22×C4)(D4×C23), (C23×C4)(D4×C23), (C22×C4)2(C22×D4), (C22×C22⋊C4)⋊35C2, (C2×C22⋊C4)⋊90C22, C22.143(C2×C4○D4), (C2×C4)2(C2×C4×D4), (C2×C4)5(C2×C4⋊C4), (C22×C4)(C2×C4×D4), (C2×C4⋊C4)(C23×C4), (C2×C4)4(C2×C22⋊C4), (C2×C4)2(C22×C4⋊C4), (C22×C4)3(C2×C4⋊C4), (C2×C22⋊C4)(C23×C4), (C23×C4)(C22×C4⋊C4), (C22×C4)3(C2×C22⋊C4), (C2×C4)2(C22×C22⋊C4), (C22×C4)2(C22×C4⋊C4), (C23×C4)(C22×C22⋊C4), (C22×C4)2(C22×C22⋊C4), SmallGroup(128,2154)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — D4×C22×C4
C1C2C22C23C24C23×C4C24×C4 — D4×C22×C4
C1C2 — D4×C22×C4
C1C23×C4 — D4×C22×C4
C1C22 — D4×C22×C4

Generators and relations for D4×C22×C4
 G = < a,b,c,d,e | a2=b2=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1756 in 1264 conjugacy classes, 772 normal (13 characteristic)
C1, C2 [×3], C2 [×12], C2 [×16], C4 [×16], C4 [×12], C22, C22 [×50], C22 [×112], C2×C4 [×68], C2×C4 [×100], D4 [×64], C23 [×71], C23 [×112], C42 [×16], C22⋊C4 [×32], C4⋊C4 [×16], C22×C4 [×78], C22×C4 [×108], C2×D4 [×112], C24, C24 [×28], C24 [×16], C2×C42 [×12], C2×C22⋊C4 [×24], C2×C4⋊C4 [×12], C4×D4 [×64], C23×C4 [×3], C23×C4 [×26], C23×C4 [×16], C22×D4 [×28], C25 [×2], C22×C42, C22×C22⋊C4 [×2], C22×C4⋊C4, C2×C4×D4 [×24], C24×C4 [×2], D4×C23, D4×C22×C4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], D4 [×8], C23 [×155], C22×C4 [×140], C2×D4 [×28], C4○D4 [×4], C24 [×31], C4×D4 [×16], C23×C4 [×30], C22×D4 [×14], C2×C4○D4 [×6], C25, C2×C4×D4 [×12], C24×C4, D4×C23, C22×C4○D4, D4×C22×C4

Smallest permutation representation of D4×C22×C4
On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 11)(6 12)(7 9)(8 10)(13 44)(14 41)(15 42)(16 43)(17 29)(18 30)(19 31)(20 32)(25 36)(26 33)(27 34)(28 35)(37 50)(38 51)(39 52)(40 49)(45 57)(46 58)(47 59)(48 60)(53 61)(54 62)(55 63)(56 64)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 63)(26 64)(27 61)(28 62)(29 41)(30 42)(31 43)(32 44)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(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)
(1 45 31 63)(2 46 32 64)(3 47 29 61)(4 48 30 62)(5 37 16 36)(6 38 13 33)(7 39 14 34)(8 40 15 35)(9 52 41 27)(10 49 42 28)(11 50 43 25)(12 51 44 26)(17 53 23 59)(18 54 24 60)(19 55 21 57)(20 56 22 58)
(1 19)(2 20)(3 17)(4 18)(5 43)(6 44)(7 41)(8 42)(9 14)(10 15)(11 16)(12 13)(21 31)(22 32)(23 29)(24 30)(25 36)(26 33)(27 34)(28 35)(37 50)(38 51)(39 52)(40 49)(45 57)(46 58)(47 59)(48 60)(53 61)(54 62)(55 63)(56 64)

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

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

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

80 conjugacy classes

class 1 2A···2O2P···2AE4A···4P4Q···4AV
order12···22···24···44···4
size11···12···21···12···2

80 irreducible representations

dim1111111122
type++++++++
imageC1C2C2C2C2C2C2C4D4C4○D4
kernelD4×C22×C4C22×C42C22×C22⋊C4C22×C4⋊C4C2×C4×D4C24×C4D4×C23C22×D4C22×C4C23
# reps112124213288

Matrix representation of D4×C22×C4 in GL5(𝔽5)

40000
01000
00400
00010
00001
,
40000
04000
00400
00010
00001
,
20000
04000
00100
00010
00001
,
10000
04000
00100
00012
00044
,
10000
04000
00100
00040
00011

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,2,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,1,0,0,0,0,1] >;

D4×C22×C4 in GAP, Magma, Sage, TeX

D_4\times C_2^2\times C_4
% in TeX

G:=Group("D4xC2^2xC4");
// GroupNames label

G:=SmallGroup(128,2154);
// by ID

G=gap.SmallGroup(128,2154);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,352]);
// Polycyclic

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

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