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G = C42.175D4order 128 = 27

157th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.175D4, C24.335C23, C23.466C24, C22.2502+ 1+4, (C2×D4).31Q8, C428C444C2, C429C429C2, C23.25(C2×Q8), C4.58(C22⋊Q8), C2.36(D43Q8), C23.Q831C2, C23.8Q869C2, (C23×C4).118C22, (C2×C42).567C22, (C22×C4).842C23, C22.317(C22×D4), C22.107(C22×Q8), (C22×D4).534C22, C2.25(C22.29C24), C24.3C22.49C2, C2.C42.202C22, C2.43(C22.26C24), C2.58(C22.47C24), (C4×C4⋊C4)⋊98C2, (C2×C4×D4).65C2, (C2×C4).360(C2×D4), (C2×C4).311(C2×Q8), C2.34(C2×C22⋊Q8), (C2×C42.C2)⋊14C2, (C2×C4).151(C4○D4), (C2×C4⋊C4).314C22, C22.342(C2×C4○D4), (C2×C22⋊C4).188C22, SmallGroup(128,1298)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.175D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.175D4
C1C23 — C42.175D4
C1C23 — C42.175D4
C1C23 — C42.175D4

Generators and relations for C42.175D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 516 in 274 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×22], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×6], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×10], C4×D4 [×4], C42.C2 [×4], C23×C4 [×2], C22×D4, C4×C4⋊C4, C428C4, C429C4, C23.8Q8 [×4], C24.3C22 [×2], C23.Q8 [×4], C2×C4×D4, C2×C42.C2, C42.175D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C22⋊Q8, C22.26C24, C22.29C24, C22.47C24 [×2], D43Q8 [×2], C42.175D4

Smallest permutation representation of C42.175D4
On 64 points
Generators in S64
(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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(1 59 51 45)(2 58 52 48)(3 57 49 47)(4 60 50 46)(5 42 62 56)(6 41 63 55)(7 44 64 54)(8 43 61 53)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(13 33 27 39)(14 36 28 38)(15 35 25 37)(16 34 26 40)
(1 7 11 38)(2 39 12 8)(3 5 9 40)(4 37 10 6)(13 18 43 48)(14 45 44 19)(15 20 41 46)(16 47 42 17)(21 36 51 64)(22 61 52 33)(23 34 49 62)(24 63 50 35)(25 30 55 60)(26 57 56 31)(27 32 53 58)(28 59 54 29)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim1111111112224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D42+ 1+4
kernelC42.175D4C4×C4⋊C4C428C4C429C4C23.8Q8C24.3C22C23.Q8C2×C4×D4C2×C42.C2C42C2×D4C2×C4C22
# reps11114241144122

Matrix representation of C42.175D4 in GL6(𝔽5)

010000
400000
003100
002200
000030
000042
,
100000
010000
001200
004400
000010
000001
,
200000
030000
004300
001100
000022
000013
,
020000
200000
002400
000300
000044
000021

G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,2,1,0,0,0,0,2,3],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,4,3,0,0,0,0,0,0,4,2,0,0,0,0,4,1] >;

C42.175D4 in GAP, Magma, Sage, TeX

C_4^2._{175}D_4
% in TeX

G:=Group("C4^2.175D4");
// GroupNames label

G:=SmallGroup(128,1298);
// by ID

G=gap.SmallGroup(128,1298);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,456,758,723,184,675,80]);
// Polycyclic

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

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