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G = C22×C4.10D4order 128 = 27

Direct product of C22 and C4.10D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C22×C4.10D4, M4(2).22C23, (C2×C4).2C24, (C23×C4).23C4, (Q8×C23).9C2, C24.126(C2×C4), (C22×C4).776D4, C4.132(C22×D4), (C22×Q8).29C4, C22.15(C23×C4), (C2×Q8).326C23, C23.221(C22×C4), (C23×C4).510C22, (C22×C4).896C23, C23.231(C22⋊C4), (C22×M4(2)).25C2, (C22×Q8).452C22, (C2×M4(2)).332C22, C4.70(C2×C22⋊C4), (C2×C4).1400(C2×D4), (C22×C4).87(C2×C4), (C2×Q8).200(C2×C4), (C2×C4).243(C22×C4), C2.29(C22×C22⋊C4), C22.79(C2×C22⋊C4), (C2×C4).280(C22⋊C4), SmallGroup(128,1618)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22×C4.10D4
C1C2C4C2×C4C22×C4C23×C4Q8×C23 — C22×C4.10D4
C1C2C22 — C22×C4.10D4
C1C23C23×C4 — C22×C4.10D4
C1C2C2C2×C4 — C22×C4.10D4

Generators and relations for C22×C4.10D4
 G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=dcd-1=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 556 in 368 conjugacy classes, 180 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4, C2×C4 [×35], C2×C4 [×24], Q8 [×32], C23, C23 [×6], C23 [×4], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×26], C22×C4 [×8], C2×Q8 [×16], C2×Q8 [×48], C24, C4.10D4 [×16], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C23×C4 [×2], C22×Q8 [×12], C22×Q8 [×8], C2×C4.10D4 [×12], C22×M4(2) [×2], Q8×C23, C22×C4.10D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C4.10D4 [×4], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4.10D4 [×6], C22×C22⋊C4, C22×C4.10D4

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim11111124
type+++++-
imageC1C2C2C2C4C4D4C4.10D4
kernelC22×C4.10D4C2×C4.10D4C22×M4(2)Q8×C23C23×C4C22×Q8C22×C4C22
# reps1122141284

Matrix representation of C22×C4.10D4 in GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000001600
00001000
00000001
000000160
,
01000000
10000000
001110000
001660000
000000160
000000016
000001600
00001000
,
01000000
160000000
001660000
001110000
000000512
0000001212
0000121200
000012500

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,11,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0] >;

C22×C4.10D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4._{10}D_4
% in TeX

G:=Group("C2^2xC4.10D4");
// GroupNames label

G:=SmallGroup(128,1618);
// by ID

G=gap.SmallGroup(128,1618);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,2804,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^4=c^2,e^2=d*c*d^-1=c^-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*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations

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