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G = C22×C4.10D4  order 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), C22.79(C2×C22⋊C4), C2.29(C22×C22⋊C4), (C2×C4).280(C22⋊C4), SmallGroup(128,1618)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22×C4.10D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C22×C4.10D4
Lower central
C1 — C2 — C22 — C22×C4.10D4
Upper central
C1 — C23 — C23×C4 — C22×C4.10D4
Jennings
C1 — C2 — C2 — C2×C4 — C22×C4.10D4

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

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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